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Chasing Lines Slowly


Learning to follow a line is a skill that will teach students to think well. This skill is mathematical, involving spatial recognition, fractional and proportional relationships, and directional measurement.

To complete this project you will need colored pencils or markers, a sheet of heavy drawing paper or a drawing journal, and a quiet space to enjoy the process.

  1. Look carefully at the drawing by Henri Matisse.
  2. Divide your drawing space in your mind into four quadrants.
  3. Choose a quadrant to draw one of the fruits that you will then anchor all the other fruits and leaves to. Be sure to consider how much space the fruit demands and where the fruit is settled in the quadrant.
  4. Swap colors and draw a shape that connects to the first shape.
  5. continue like this until the drawing is complete.

Remember, teasing students that good things take time is an important lesson. This lovely drawing was completed by a 3rd grader in three sittings.

Ready, set, go (SLOWLY)!



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Fibonacci on the Produce Aisle

Math is everywhere. Absolutely everywhere!

When I saw this vegetable ate the market, I stopped to ogle. It was fun to learn that it is sometimes called Romanesco cauliflower, sometimes Romansch broccoli. Either way, I was not thinking of the thing as food, only sheer math! 

Can you see it?



And then I thought to myself, "This is math my students can get behind." So I pulled out a book:


And later I stirred up some soup and called my family to dinner.



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At the Intersection of Math and Writing



I began the writing workshop with Cuisinare rods and colored pencils. My writers looked puzzled.

“Today you're going to make a Cuisinaire construction and then describe how you made the construction with words on paper so that a reader will be able to navigate through the paragraph to create an identical construction.”

This is my idea of a hands-on How To paragraph.

“Just like math, when writing instructions you have to show all your steps.”

Young writers need to practice working through the process of crafting words. It's challenging teaching young writers that words need to be wisely chosen and crafted carefully to accurately communicate a specific idea to an audience of readers. This is challenging because the task is a process that involves tremendous effort on the part of the writer and young writers want to skip steps. Participating in this work over time sets a foundation for the rhythm of the writer's routine to be established.

Before beginning, I  challenged my writers to keep in mind the cardinal rule of our writing workshop:
“Words are scribbled on paper for a reader to read… your words are a gift.”

The young writers eagerly spent an hour an a half contentedly drafting rough drafts paragraph that they took home to self-conference and craft to final draft.

“Next week we will exchange final drafts and see if readers can make the construction.” 

Begin all writing experiences by breaking the task at hand into steps. Remind writers that writing is a process. Getting young writers to engage in process is a tricky business that takes time to root, but truth be told, process alone takes the daunting out of writing.

We broke this specific project down as follows:

What's your big idea?
Make a construction with Cuisinaire rods. Map the construction on graph paper with colored pencils.

Write it down…!
Begin by use a topic wheel to outline each step involved in the construction. Craft a paragraph following the topic wheel outline. Be sure to introduce the topic with a sentence that hooks the reader into the big idea. The supporting sentences should include specific details that will allow the reader to navigate through the Cuisinaire construction without a hitch. Craft a single sentence at the end of the paragraph that will conclude the exercise and add an interesting clincher that makes idea of the paragraph echo in the mind of the reader.

Conference with yourself and someone else…
Now, re-read what you wrote and decide, as a reader, if you are accurately communicating your big idea. Use a red pen to make changes. Ask someone else to read your work and to add red marks when they find confusing areas, holes, or dead ends in your “How To” paragraph.
Make a final sweep with the red pen for common errors—spelling, punctuation, capitalization, tense, and so on.

Final Draft
Use your best handwriting or type up a final draft!

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“There are several words that sang above the rest in my high school science classes.

From botany, photosynthesis is the one.

From marine biology, Echinodermata and Coelenterata.

From chemistry, stoichiometry.

And, from cellular biology, mitochondria captured my imagination.”

So begins the lesson.

"Mitochondria located in the cytoplasm are little energy factories within the cell. These amazing organelles enable respiration, which allows the cell to move, to divide, and to thrust their unique purpose. Mitochondria can have different shapes depending on the cell type. Because they contain their own DNA, ribosomes and can produce their own protein, mitochondria are only partially dependant upon the host cell."

What I set out to explore with my students is the fact that mitochondria possess a double membrane, an outer, which is smooth, and an inner, which possesses many folds called cristae which exponentially increase membrane surface area.

“All living cells have mitochondria. But it is amazing to consider that typical animal cells have up to 2000 mitochondria… in each cell!”

I wanted to take their imagination on a journey between these folds.


“Folds give mitochondria their unique potential; enable the organelle to be highly productive. Cristae take batches of sugar and oxygen and produce ATP (adenosine triphosphate)—cell food.”

At the beginning of the year, science began by exploring the idea that science and art are uniquely connected.

Leonardo himself reminds us, “All our knowledge has its origins in our perceptions.”

My hope was to connect the exploration of mitochondria to the unit we had just completed on the human nervous system. We explored the potential of individuality as we explored the brain—human potential, genius. And here another potential to bridge the gap between learning information and sparking individuality presented itself, this time on a cellular level.

“So today, to continue our exploration of mitochondria, we are going to watch a film about origami.”

Yes, origami.

The students gathered round the TV. I popped in the DVD and set out to accomplish some administrative goals.

Not far into the film I overheard the little group letting out amazement. I was not surprised. But soon I witnessed something that caught me off guard. One-by-one individual students from the group ranging from the 5th through the 11th grader, got up to grab a stack of paper.

They were folding.

The film did not provide a directive to viewers. This was not a "fold-along" film. These students were engaging in the task spontaneously.

Being inspired is magnificent.

During the next biology workshop I provided instructions and large pre-cut squares of paper for the students to fold a hyperbolic parabola. This, to reinforce the film’s message that even paper has hidden potential.

“Folding paper is work. But your work is not in vain. Your work utilizes a fraction of potential. And the paper will never be the same.”

Dare I say, neither will they?

I think mitochondria is one of those words that will stick.


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Happy New Year: Here Goes, Math

I am squirming, back-to-work-Monday, the first of the year smirking around the corner. Let’s face it, even teachers who love teaching have to oil their gears after three weeks of fa la la.

So I find myself clicking through my past philosophical musings trying to remind myself why, exactly, I chose this profession until I stumble on an apropos reminder, “The comfort of routine, once established, will set roots deep into soil, establishing a framework for the tree to grow strong. When a routine rhythm is established from an early age, the student will value the work of exploring…” That’s the one!

To value the work of exploring, now there’s a worthy goal.

Math is a subject where consistency is a must. Fine-tuning a math routine is fingernails on chalkboard, a nearly archaic metaphor to be sure. Seeing as chalkboards have gone the way of record players, here’s an opportunity for a few seconds of Youtube diversion:


Okay, back to math. Math is a keystone subject that presents bumpy stretches of road along the way. Challenging our students to do their math consistently will give them the ability to be successful.

But what else can be acquired on the journey?

I learned a long time ago that, like all subjects to be tackled, a systematic approach to math will not only enable students to strengthen their math skills but will allow them to experience the discipline of working through a process to accomplish a task, and this, this, my friends is more valuable than the actual subject being tackled. "Process" after all, is the key to writing, literary analysis, visual art, music, and historical research. The list goes on and on and on.

What do I want to see in the classroom this year?

Students totally immersed in their learning, students discovering their individual efficacy.

And so I nod to myself, yes, that’s why I venture into unfamiliar territory, that’s why I take a whole-brain approach to solving problems, yes, whole-brain even when it comes to math problems.

Numbers and number relations, fractions, patterns and functions, data analysis, probability, algebra, no matter the strand, my goal is to help my students to move from simply completing a math lesson or math exam with exceptional accuracy to something much more.

Each of my students are plugged into a traditional math textbook and set on an individualized journey. I currently utilize Teaching Textbooks or Saxon depending on the individual needs of the student because of the exceptional didactic element that, if utilized over time, provides a potential for students to own their study of math. Beyond that, I am bent on incorporating concrete instructional materials that allow my students to delve into an exploration of critical and creative thinking. Students who engage in concrete learning are better able to apply what they’ve learned in real life situations. Fact of the matter is that students who use concrete materials develop more precise and more comprehensive mental representations—especially math students. Often times that these students are more motivated because experience with concrete materials have enabled them to develop longer attention spans. The benefits are endless.

Learning that moves through stages is learning that sticks. Students should begin in the Concrete or “doing” stage of learning because it enables new ideas to connect with familiar ideas. Building conceptual understanding of this nature supports retention, prevents common errors, and allows students to make larger critical and creative connections. From here students will move with ease to the Representational or “seeing” stage of learning, transforming the concrete into visual representations or pictograms. Moving through these first two stages often eliminates “holes” in mathematical understanding and allows students to confidently reach into the Abstract or “symbolic” stage of learning. Once arrived, the capacity for logic, for reflection, has blossomed to the point that the student begins to believe in the diligence that makes them hungry for more.

I have been striving for years to perfect my vision of a “Math Lab” upon which to found the math textbook. Funny, this year the goal is creeping to the top of my list.


Passionate learning, of course!

Here’s to more of that in 2011.

– Kim