Category: Blog

Math as a Laboratory Science

Math is not only the last letter in STEM or STEAM, it is the only one that we do not picture as experimental. We don’t imagine students learning science without doing experiments. We don’t imagine them learning technology without writing code, or learning engineering without building models, or learning art without messing with paint, clay, or paper. Yet, we easily imagine learning math without experimenting. In fact, it is rare that students ever do a math experiment or think about math that does not have a “right” answer.

I learned to experiment from one of my great teachers, Walt Hunter. I even had the great good fortune to also being his chemistry lab assistant my senior year in high school. That I did not fall in love with chemistry was not his fault; I had just loved physics since I was 7 years old. But I did fall in love with experimentation, and like Walt I gained a deep belief that learning to experiment should be an essential aspect of every student’s education. I brought that belief to my physics classes replacing teacher demonstration with student experimentation. I took it to my Jr. High math classes, where I made my students worksheets that let them play with numbers and mathematical patterns. I carried it to my focus on manipulatives as a math coordinator, and I bring it to What if Math.

Using spreadsheets as basic learning tools for math has many advantages, but I think the most important one is that it turns math into a laboratory science. It enables students to experiment, to build and iterate models, to test those models, and to apply them to real-world data, complex rich data. It lets them ask and answer what if… questions. And it turns them into explorers who love to use math and who gain Walt’s experimental habits of mind, the thrill of discovery. It is this, I now know, that Lynn Steen saw when he described mathematics as the “Science of Patterns,” for math does belong to STEM/STEAM after all. So, when you plan your math classes, imagine your chemistry teacher, and the twice weekly labs where you learned to act like a scientist, to explore, to discover, to ask, “What if…”

Art

*Portrait of Antoine-Laurent Lavoisier and his wife by Jacques-Louis David, ca. 1788, Wikipedia

This is Why I Love Graphs!

This graph appeared on one of my favorite websites – Statista.

Given the “breaking news” of the day, that the President wants to impose new tariffs on steel imports, it is fascinating to see from this graph the countries most affected, certainly not the ones we might have thought. It is a perfect example of the power of visualization, of graphs, to tell a story, and the reason we consider them fundamental to our What if Math Labs. Take the Tour to see more.

Art

Revolutionary Math

Cape Cod in the winter is one of those marvelous places filled with interesting shops and people waiting in the quiet winter time for the soon to come crowds. It was on one of those pretend spring is here days in February that we went to visit a dear friend on the Cape and then take a lovely drive to empty beaches, delicious lobster rolls, and of course a bookstore or two. It was on that last stop, just before the bridge, that I came upon a hidden treasure, a math textbook from 1788. The author Nicolas Pike entitled it, A New and Complete System of Arithmetic: Composed for the Use of the Citizens of the United States. Pike, proud of his brand new nation so recently created, says that it needed a book to educate its newly minted citizens in mathematics.

It is however the opinion of not a few, who are conspicuous for their knowledge in the mathematics, that the books, now in use among us, are generally deficient in illustration and application of the rules; of the truth of which, the general complaint among schoolmasters is a strong confirmation….as the United States are now an independent nation, it was judged that a system might be calculated more suitable to our meridian, than those heretofore published.
Pikes Arithmetic, Nicholas Pike, 1788, Preface.

The book follows, for the most part, the sequence and topics laid out by Leonardo of Pisa in Liber abbaci. Though it lacks pictures, it is full of contemporary problems, problems faced by farmers, shopkeepers, traders, surveyors, sailors, and even militia. It is full of such real-world problems and full of tables to help the users to calculate the answers to those problems. It was a reference book as well as a textbook. It was designed for this new country, “suitable to our meridian” including decimal currency.

Over the past 230 we have desiccated this work, taking out its focus on problem solving in the real world, both in the problems given and in the tools for solving them. I love the thought that in What if Math we are returning to Nicholas Pike’s 1788 vision, to focus school learning on the kinds of problems students will need to solve and giving them training in the tools and skills they will need to use. Yes, this was a revolutionary vision then and it is a revolutionary vision today. But it is a vision for a nation whose promise has been: to enable all of its citizens to thrive. I would love to have been able to take Nicholas Pike on our Tour. I think he would have liked it.

Art

The Bit

The key to the digital age is also the key to learning algebra.

Despite what many of us may believe, our digital age did not began with the microprocessor, or the personal computer, or even the iPhone; it began with a single amazingly simple idea by a quiet man who few of us would today recognize. Claude Shannon grew up in Minnesota when radio was becoming the means of communication to all, broad cast. It was the age when sound was added to movies, when phonographs and records storing sound became a must in every home, when the first facsimile machines were used to transmit photographs and text, and when everyone could take their own pictures with the Kodak Brownie camera.

Each of these transforming inventions used a different analog means of storing or transmitting data. Analog data is continuous; on a graph it is a line, sometimes smooth, sometimes jagged. All of these inventions had to deal with the problem of noisy data and of separating the noise from the data. This was the problem Claude chose to work on. Before him the common way of dealing with noisy data was to turn up the volume. If the radio static was bad, make it louder. If the picture was muddy, increase its contrast. If the telephone call was hard to understand, yell.

To solve this problem of noisy data both in storage and in transmission, Claude came up with a truly brilliant, surprising, and original idea. Think about all data as digital. Think about it as being broken down into discrete bits, a collection of just 1’s and 0’s. No longer would data be stored or transmitted as a wave like the grooves in a phonograph record, a continuous quantity. In Claude’s new world it would be like atoms, discrete, separate, objects. Bits, the word he chose, came from binary digits; where his “atoms” took two and only two forms. It was transmitted in bits, stored in bits, and processed in the same bits. He then figured out how to find corrupted noisy data, how to minimize it, and how to replace it. When he died at the turn of this century, his vision for data was just becoming an overwhelming reality. Because today, we have the bandwidth, the storage, and the processing power to handle all data digitally, and the processes that make noise no longer a problem we concern ourselves with.

Isn’t it time our schools deal with its noise problem by becoming digital and focusing on discrete data? Today’s “analog” continuous variable algebra makes the concept of variable abstract and difficult for many students to understand. It requires students to learn a complex set of special cases to solve abstract equations. It turns algebra into collection of mechanical processes focused on cases that are easy to solve. What if we were to follow Claude Shannon’s lead and treat variables as discrete, digital quantities? Spreadsheets make this easy. Variables become concrete, easy to understand, iterate, build into functions, and use those functions to build models. They give us the means to focus on real, messy, interesting data to solve fascinating problems.

Try this new way of thinking for yourself. Go to our Tour to see apply the digital world to algebra. Try it with your students. Tell us what you think.

Art

Exhausted

Teaching done right has always been a hard job, but it is now substantially harder. Talk to any teacher and they will tell you that they are overwhelmed. Blame it on kids more distracted, on parents more demanding, on the misery of an over reliance on testing that saps creativity and judges teachers on things they cannot control, on a lack of money, on cell phones. The list is endless, personal, and the results exhausting.

If we seek not blame but instead deep cause, we will see that much of the pain teachers are now rightly feeling is due to the new digital technology, technology that has had a positive affect on most other aspects of our lives. Digital technology in the form of cell phones not only distracts students, it invades teachers’ lives, for they feel the need to answer students queries 24/7. Email which has become a primary form of communication opens the door to parent-teacher and student-teacher dialog again extending the school day and adding burdensome demands. Powerful computers now enable standardized testers to analyze data and grade teachers on student progress. Shared syllabi on common instructional platforms rigidly sequence and control teacher lessons removing any opportunity for creativity and innovation. The scope and sequence that used to weight down teacher desks collecting dust in the bottom right hand drawer are now online controlling the day.

Word processors, while making it easier for teachers to read written work, also make it easier for students to write more and to demand that teachers immediately read, respond to, and grade it. PowerPoint presentations of content are not as easily erased as chalkboards, saving class time but demanding more preparation time. And like doctors today, teachers too, feel the need to be up on the latest info available on the Web. Last year’s lecture notes just won’t do any more.

New technologies can be insidious. While as teachers we may worry about big tech issues like flipped classrooms, online assignments and tests, personalization, and the need to ensure our students have equitable tech treatment; we must also prepare for the future of blended classrooms and online courses. Digital technology has made our lives harder, much harder.

Technology is always like that. It starts out by making us work harder. It requires us to follow a learning curve. It demands we learn new ways to do old things that do not make them easier or save us time. And it is invasive, causing us to add new problems like student security to all the old problems. This is where most teachers at both K-12 and college find themselves, fighting digital technology instead of enjoying its benefits. For powerful new technologies, technologies that change our lives, require us to not just adapt our old forms, methods, and content but to rethink them. It requires us to learn to fully use technology and integrate it with what we are doing. And it requires us to imagine our role in a new way.

What can you do to make technology work for you and for your students? What can you do to make your job easier instead of harder? What can you do to prepare your students for a world you were not prepared for? Stay tuned!