Category: Blog

Change

“Today, it seems as if nearly everyone agrees that high school mathematics needs to change. For far too long high school mathematics has not worked for far too many students: too many students leave high school unprepared for college or a career, particularly a STEM career; too many students do not see how math is useful in their lives; too many students leave high school without an affinity for doing math; too many students leave high school without the quantitative skills necessary to make sound decisions in their personal life and in our society which is increasingly quantitative in nature.  High school mathematics has not changed substantially in my lifetime, nor has it changed substantially for most students, teachers, schools, districts, and states.  It is clearly an issue—and it is a critical issue of access, opportunity, and equity.” By Matt Larson, NCTM President
October 25, 2016

We are thrilled that NCTM agrees that the math we teach is obsolete and does not serve our children. We hope you find the great new stuff that we have coming very, very soon, does.

Art

Back- to-School – add 10%

It was forty years ago this September that I started my career as a high school mathematics teacher, a career that spanned 36+ continuous years. I continued some teaching and other classroom work over the last few years, but this will be the first September I do not find myself preparing to greet some new students as their teacher. Since the school calendar has guided the activities of my life for as long as I can remember, I anticipate that this time of year will always signify a beginning for me, and I imagine I will always reflect on Septembers of the past, and I will always be thinking of the students and educators embarking on a new year of learning and growing.

The discounts offered in this year’s back to school sales remind me of a September years ago that influenced my back to school routine. My supervisor, who I had worked with for many years, always did his best to motivate us with new ideas for a successful year. One year his message was a challenge – change 10% of your practice this year. He offered some thoughts, sent us off, and I did my best every year to implement something new or different, just a manageable 10%.

Technology was an area of my frequent ‘changes’. I found instructional technology a powerful tool for engaging students to explore mathematical concepts, for differentiating learning experiences, for challenging students in ways that would provide immediate feedback to them. In a world where technology drives rapid change in so many areas of our lives, it seems important to consider how we might all seek to use technology to change 10% of our practice this year!

Peter Mili

What Algebra?

Each summer, as schools get ready for a new school year, the question returns, “Should we be teaching algebra to our children?” it seems to have been started by Andrew Hacker who has tried to argue and continues to argue that algebra is both difficult and an unnecessary burden for most of our students, and thus should be dispensed with. He continues to ask and the echo continues to resound, but they focus on the wrong question. The question should not be “Whether algebra?” but instead must be “What algebra?” For there are many algebras, but in particular we should focus on two, the algebra of solving equations and the algebra of functions.

Most of the algebra we teach our students is the algebra of solving equations for unknown quantities. Its origin dates back almost 1300 years to the Muslim flowering. It is the algebra we are all familiar with using x as the sign for the unknown quantity and the collection of techniques for finding out what x is equal to. Its fundamental methods, balancing and completion, are found in the title of the great work from Al Khwarizmi from which its name derives. Today it has become little more than the practice of manipulating symbols for the problems that it was designed to solve are no longer the primary problems of the business community which stimulated its practice.

The other algebra that we trace directly back only 500 years to the late 1630’s is the algebra of functions. This algebra does not solve equations, it builds tables and graphs; it is not about unknown quantities, it is about variables, quantities that can and do take on multiple values; it relies not on a series of paper algorithms, but on dynamic engines that graph and change. It is, in sum, not a static algebra but a dynamic algebra and thus the algebra of science. And of most importance, it is the algebra of spreadsheets making it today, the algebra of business too.

The algebra of functions need not and does not rely on the algebra of equations as a foundation, particularly in this time of computers where spreadsheets automatically compute. It does not need the mechanical skills of solving specialized forms of equations, nor the conceptual confusion of this x thing called by the algebra of equations “a variable” but in reality a placeholder for an unknown number. Introduced in schools today as a modest extension of the old algebra for most students, the algebra of functions only makes a serious appearance for those planning to take calculus. Despite its importance in business today, the algebra of functions and their spreadsheet tools are not a significant part of our standard math curriculum even in college algebra courses where the paper and pencil mechanics of functions are studied but not, not their applications. It is no wonder that our students complain so bitterly that the math they are forced to learn is abstract, meaningless, and utterly useless.

We should not be asking whether we should be teaching algebra, we should be asking “What algebra should our students be learning?”

To learn more you may want to read my paper, The Idea that Changed the World.

Mastery

The word seems so benign. Yet it has become the goto word in education. School superintendents, even the best and most advanced of them, use it all the time. “We want each of our students to reach mastery in each of the standards.” To master something is to be in command of it, to control it, to know it. It seems so logical and natural to want every student to achieve mastery that we now make it a synonym for learning, the product of schooling which we can objectively define and measure.

But if learning is to be a creative experience, how can it be mastered. Creativity by its very nature is constantly evolving, constantly pushing us to think out-of-the-box. Can we really learn to master creativity?

Apple is trying to do just that with its new headquarters in Cupertino, CA. It is seeking to make innovation a product that can be engineered, that can be structured, that can be mastered. Steve Jobs started its design, and Tim Cook and his leadership team is finishing it for Steve. It is a huge building, the size of several Pentagons, built to bring together in one place much of Apple’s creative and product development workers. It is designed to enable Apple retain its innovation leadership as well as its status as the largest company on earth. Laid out in a giant torus to bring people together to share ideas and share visions in both formal and informal opportunities, it’s goal is to master and manufacture creativity.

If Apple is right and a building designed to express and indeed enhance a corporate culture of creativity can make creativity a product, then mastery in schooling is right. If they can master creativity, build products that are inherently, creative then Apple will retain its competitive edge. If we can teach students to master concepts and thus master creativity, then a focus on mastery must assuredly be the goal of education. But if creativity is a process, a process of continuous growth and not mastery, a process of practice in which all we learn is to always ask “What if…” then creativity can never can be mastered or finished or finally learned. For it is a process of continuous growth, exploration, and experimentation, a process we become increasingly fluent in. If creativity is a process and not a product if our goal is fluency and not mastery then Apple should rethink its reliance on architecture and education should rethink its goal of mastery, for neither will produce the creative products or learning we seek for our future generations. Our goal must be creative fluency and not content mastery.

The End

Despite the many attempts to codify the creative process, it is as surprisingly individualistic as it is human. John Irving, author of iconic works like The Cider House Rules, describes his creative process as writing the conclusion before the beginning. He spends a year or more developing a story, the plot, and a set of characters in his head, writing nothing down, until he has the whole in mind. Then he writes the final paragraph. Then and only then does he start actually putting words down on paper going from beginning to end. His description of the writing process is a fascinating view into the creative act, most fascinating for me because I do it so differently.

I come up with an idea, often in the quiet mind time of the morning shower or the last thoughts as I slow my mind on the way to sleep. It is usually a blurb of an idea, often captured in a few words or a sentence or two. Never more than that. I repeat it over and over again to myself so that I will remember it, because my memory is notoriously terrible. I sometimes work on that idea for a few days, coming back to it, testing whether I still like it, adding a few connections to it, but rarely carrying it much further. I basically write that sentence of two on my brain and hold it there until I have the inclination or the time to actually develop it, compose it. For I am most fascinated not with the original idea but with what will become of it, where it will go, what it will develop into when I start keying it in for real. To me, the thrill is in discovery. It is in running the experiment and finding the result. It is in seeing the working invention. It is in printing the beautiful picture that I have tweaked and messed with. So I find writing not a painful demanding activity but a creative thrilling one that I take joy in. For I never quite know where it will go or what it will produce. I suppose it is the reason I so love asking “What if…” in our spreadsheets. What does your creative process look like? I just learned more about mine!