Monday, October 21, 2013



A square is a rectangle. This is the statement that I found myself explaining -- actually passionately defending -- with my 6-year old (Lizzie) and 8-year-old (Anna Mae) a few nights ago when straying off topic from Lizzie’s math homework. A rectangle has four sides and four right angles. So, any polygon (or using Lizzie’s language: a shape that has no holes and no curvy sides) with exactly four sides and four right angles must be a rectangle. Ergo, a square is rectangle. Ladies and gentleman of the jury, I rest my case.

I know what you’re thinking. No one likes a math nerd (especially one that pretends he’s Matlock) and isn’t this just some technical nuance of language that will only confuse my daughters that are simply learning the properties of basic shapes? Sadly, I have to concede: you’re probably right on all accounts. But, really it’s a lesson that goes deeper than geometric properties. It’s a lesson on mathematical reasoning, on problem solving, on logic - on starting with a few rules or assumptions and building a whole world of conclusions. So when I saw the window of opportunity for spreading my love of logic to my daughters open slightly, I dove through head first.

Tucked away in your head somewhere is the memory of learning about postulates and theorems in high school geometry. Postulates are the basic assumptions which are simply accepted as true. For example, postulate one states that a straight line segment can be drawn joining any two points. This very same postulate was scribbled on some papyrus by Old Man Euclid in 300 BC. (He probably preferred to just be called Euclid back then.) Euclid also included four other postulates and five axioms or “common notions.” The basic truths or assumptions are the seeds from which enormous mathematical forests grew. Using only a few basic assumptions, philosophers and mathematicians were able to conclude a few more things, which allowed them to prove something else, which in turn allowed us to determine another thing, opening the door for the next generation to form the next conclusion, and so on. It’s like a Rube Goldberg machine that just keeps propelling the next act. A marble rolls down a ramp that strikes a domino which falls onto a switch that lights a match that causes air to fill a balloon… Only the mathematical machine never comes to an end - with each stunt becoming more complex than the one before.

If you’ve ever even stacked dominos in a row, you know that each domino is completely reliant on the one that precedes it. The same is true of logic. Place one domino a little out of position and the dominos that follow will remain standing. Introduce faulty logic along your path of mathematical discovery and everything that follows is erroneous. This is why I love math and why I want my girls love to math. For me, it’s never been about algorithms or contrived steps that march you from point A to D. It’s about knowing that you have a bag of tools - tricks really - that if applied accurately and appropriately, allow you to discover your next destination. Mathematicians don’t just solve problems; they uncover opportunities.

Reasoning and logic also teaches us of the importance of knowing that the beliefs that we hold are rooted in some basic assumptions that we each individually hold to be true. Time and time again, we’ve learned that what seems obvious and certain based on our individual perspectives do not fit the perspective of others, the believes of the next generation, or even the facts that we’ve yet to discover.

Galileo and a few other scientists before him had the courage to question that the Earth was the center of the universe. Albert Einstein challenged some of Isaac Newton’s assumptions and showed that of Newton’s Laws of Motion were only approximately correct, falling apart when objects approached the speed of light. Euclid’s very own fifth postulate became quite the controversial topic for nearly 20 centuries. (You know how mathematicians like to find fodder.) The postulate said something like: “At most one line can be drawn through any point not on a given line parallel to the given line in a plane.” There was much debate about whether or not this really needed to be included as a postulate. It wasn’t that mathematicians necessarily thought of it as a concept that didn’t hold true, it’s just that they thought it was really unneeded; that his other basic assumptions (or postulates) basically had him covered. (In modern time, think of it as using unnecessary or redundant lines of code resulting in bloated, slower software.) Well, they were wrong. In fact, the more they tried to prove that it wasn’t needed, the more interesting things became. It was finally around the 19th century that entire branches of geometry - that use a different fifth postulate - became accepted as plausible alternatives giving shape to non-Euclidean geometries like elliptical geometry and hyperbolic geometry.

So, logic teaches us that a square is a rectangle. It also teaches us animals in the air likely have wings and that it’s wise to wear a jacket when we see snow on the ground. But, logic also teaches us that our conclusions are based on some definitions or assumptions. Logic reminds us that in every argument or nearly everything we hold to be true, there are assumptions. Mathematics and logic teaches us not just on how to build on our assumptions to form a stronger argument; it forces us to acknowledge the vulnerabilities of our thoughts and respect the positions of others. And, that is a lesson I want my kids to know.

Saturday, October 12, 2013


I thought that it was only fitting to begin this blog with the first article that I wrote in our district newsletter outlining the case of why we were beginning an iPad 1:1 implementation in 2010. The article attempts to highlight a few technological advances that are changing the skills required of our students and the importance of empowering students to chase their curiosity. Likewise, the article provides a quick backdrop of some of my beliefs that shape the theme of this blog - to encourage all of us (not just students) to keep finding new questions and better answers.

June 2010: Why Change?
My oldest daughter, Anna, will be starting kindergarten this year. Over the last couple of years, like most curious preschoolers, Anna has asked me an endless list of questions: “How far is the Sun from the Earth? Why do bees sting? If one giant stood on another giant, could the top one touch the moon? How do you make ice cream? What is war? Why are fireworks so loud? What causes cancer?” I’m sure every parent has experienced the insatiable curiosity of their own child. I still remember searching the library card catalog as a child and flipping through microfiche as a college student to find answers to my own questions. Anna, on the other hand, will never even hear the term “card catalog” or “microfiche.” To help answer many of her questions, she and I turned to Google and various websites. A post on Facebook and a link to a YouTube video guided us through how to make homemade ice cream. A Skype conversation with my dad who served in the military helped answer the question of war for my daughter. Unfortunately, her desire to know the cause of cancer currently is still unfilled.

These experiences remind me of the natural curiosity that all kids possess and that the tools that my daughter will use in her schooling will be – should be – very different than the tools that I used two decades ago when I graduated from high school. She, and her generation, will need to be better equipped to use tools like Google, YouTube, and yes, even Facebook, effectively and responsibly. Already today, our students live in a digitally-connected world outside of school that includes texting, social networking, virtual realities, and online games.

Google is currently attempting to scan and digitize more than 50 million books from five of the largest libraries in the world. Amazon is now selling more electronic books than hardcover versions. Apple iPad owners have downloaded over five million books in just the last two months. Facebook now has more active accounts than the entire US population. Approximately 24 hours of video footage is being uploaded to YouTube every minute. Just imagine the changes that Anna and her classmates, the class of 2023, will see during the next 13 years.

These changes surrounding our children will certainly require them to have a different set of skills than what was required just a decade ago. Anna’s future boss may not expect her to know the capital of Maine, but will expect her to know how to quickly find the answer. Our future graduates must not be equipped with just the three R’s, but must be equipped with 21st-century skills of problem solving, critical thinking, communication, and technological literacy – skills that don’t necessarily come from reading a chapter in their current textbooks. They will need to be able to quickly and accurately find answers to questions, synthesize information, communicate and collaborate with colleagues not just in their own office, but with a global community of colleagues and customers.

We as educators, parents, and community members need to acknowledge that these forces and changes are real and embrace the challenge of helping students responsibly navigate their positive uses. With this in mind, our district is trying to make a transformation to “Digital Instruction,” in which each student is issued some type of Internet device, such as a small laptop, netbook, iPad, or digital tablet. During the 2009-10 school year, we piloted this idea with a handful of classes (one in grade 6; the others at the high school). Over the course of the next year, with the help of parents and community members, our district will investigate funding sources and the details and logistics of how to make this a reality for more of our students.

Admittedly, this initiative won’t be easy. There is no denying the scope of this project and the challenges that it presents. This will require a change in policies, habits, and methods by everyone. There will be mistakes, growing pains, and adjustments that need to be made. Teachers, administrators, parents, and community members will need to work together to make it a success. I know there will be some that believe that the challenges and risks of such an endeavor are too great. I contend that the risks of not embarking on this journey are even greater.

As Anna begins school this year, I have tried to brace myself for the inevitable passing of time that will likely tick by as quickly as the last five years. Although I know her interests and personality are bound to change over the course of the next 13 years, I hope she keeps her curiosity and hunger for learning. That desire to learn and the ability to think and learn on their own are perhaps the greatest gifts we can give our students. Teachers and parents will still be the most influential force in sparking that curiosity. The device is just another tool of learning – one that helps students find their answers. But, providing this device to students opens the doors to powerful, new opportunities. If together we can figure this out and keep our students inquisitive and searching for answers, I know they will not just be successful in school, but in life. Who knows - maybe someday one of them can solve the question about cancer that Anna’s daddy wasn't able to answer.


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