Category: Explorations

Is the Textbook Dead?

It caught my eye, this headline/story posted on EdWeek recently. Seems there was a panel at a conference that was supposed to debate what they obviously thought would be an attention grabbing, contentious, and controversial topic. Their conclusion: NO!

All I can say is: “You have got to be kidding!”

Now, I know that textbooks continue to play a central role in most of our schools across the grade levels. I know they have done so for centuries, I collect antique math textbooks. And I know that both the textbook publishing community as well as the school community believes that paper textbooks will slowly morph into online interactive versions. Textbooks are so ubiquitous, so standardized, so traditional that most of us cannot imagine school without them. So is it any wonder the panel came to its conclusion: the textbook, designed for print on paper (text is derived from the Latin for tissue) will always be with us. And while many expect paper to morph into tablets, few imagine fundamental change in form. So, I continue to ask: “You have got to be kidding!”

In 1962 by Thomas Kuhn published a revolutionary work called The Structure of Scientific Revolutions introducing the term paradigm into our lexicon. Kuhn argued that science changes for the most part continuously “normal science”, but the history of science is punctuated (to use Stephen Jay Gould’s term) with “revolutionary science.” We need only note the Copernican Revolution, the Newtonian Revolution, Maxwell’s Field Theory, Einstein’s Relativity, and Quantum Mechanics in the world of physics. These scientific revolutions introduce new paradigms, fundamentally new ways of thinking that change the focus and direction of a scientific field.

I would argue that technology, like science, grows in the same way. Most of the time it normally grows by small incremental changes, improvements, but every once-in-a-while its history is punctuated by revolutionary changes. The iPhone was not a mere smaller version of a corded or a better cord-free phone. It was a fundamentally new experience, a transformative experience that changed the ways we communicate. The integrated circuit changed the way we work. The Web revolutionized the way we learn.

So, the textbook, as we know it, an invention enabled by cheap printing in the middle of the 19th century, provided a way for large numbers of students to “take a teacher home.” Its lack of interactivity meant it did not replace a teacher, but for perhaps half of the student population it provided an effective supplement of class time with additional practice and information. It was not so much a tool for learning as a tool for practicing what you hopefully learned in class.

Digital technology with its amazing interactivity, its dynamic communication capacity, and its opportunities for collaboration, gives students powerful tools for learning. What if… we could use digital age technology to enable students to learn on their own without the direct instruction of a live teacher? What if… the new paradigm for the student’s learning tools was not dependent on text or repetitive mind-numbing practice? What if… we placed, into every student’s hands, the interactive power of the Web to imagine all learning as a science experiment.

At What if Math, over the past several years, we have been reimagining a math education, indeed a STEM education, designed for the digital age. Over the past several months we have made substantial changes to our content and our website as we have come to understand learning in the digital age. Over the next several weeks we will be rolling out the last of these changes.

This is not a new textbook for the digital age. The textbook is dead. This is a new way for students to learn. It is the way we believe, someday in the not too distant future, all students will learn. We look forward to your thoughts.

Art

Touching the Sun

The Parker Solar Probe was launched yesterday to study the sun. Sixty years ago, Eugene Parker launched my scientific career. A young physics research scientist at the University of Chicago, Parker volunteered to be a mentor to encourage science-promise high school students to develop their own science projects. I had the awesome fortune to be his mentee. My project, to develop a fuel cell, may have been outside of his focus on solar wind, and I may have not understood his fascinating story, but I remember both his kindness and scientific attitude that so excited me then and animates me to this day. Of all the things I did in high school, this project stands out in my memory. It did not even work, indeed it was a complete failure, I never got any current from that battery, but with this project I began to learn the “scientific attitude” that remains the core of my work and life.

That was the first time I did real science, not a high school subject, not copying something out of a book. I was doing something new, seeking to discover, to explore, to invent. Parker was patient with this very young and immature scientist, procuring sintered graphite and other materials, asking questions, suggesting things to read, and probing, always probing. I had always wanted to be a scientist, so I do not credit Eugene Parker for putting me on that path, but I do believe that his gentle nudges introduced me to the scientific habits of mind I carry with me today, the spirit of discovery, the thrill of invention.

He is remembered in this spacecraft and in the scientific community for his discovery of the solar wind. He is remembered by me for making science and scientific problem solving — real. I thank you Mr. Parker for giving me your time and your gift. In my work I seek to bring it to all students bathed by our great sun, and by it make problem solving real for them too. Hopefully they will look at the video and fall in love as I did.

A Book or a Course

I have long loved Maxwell’s Equations as the epitome of beauty in physics and as the source of inspiration for my teaching. But though the equations are beautiful and even familiar, very few people understand them. So, when I came across this paper by the great physicist Freeman Dyson called “<a href=”http://www.damtp.cam.ac.uk/user/tong/em/dyson.pdf” rel=”noopener” target=”_blank”>Why is Maxwell’s Theory too hard to understand</a>?” I could not resist reading it. His telling of the Maxwell Equations’ story led me in a new direction not just in thinking not about physics but about education in the digital age. It led me to ask: “What’s the difference between a book and a course today?” and to further ask, “What will they look like in the future?” Before you help me tackle those questions, I suggest you look at the story Dyson tells about Maxwell’s great work.
<p style=”padding-left: 30px;”><span style=”font-size: 14px;”><em>In the year 1865, James Clerk Maxwell published his paper “A dynamical theory of the electromagnetic field” in the Philosophical Transactions of the Royal Society. He was then thirty-four years old. We, with the advantage of hindsight, can see clearly that Maxwell’s paper was the most important event of the nineteenth century in the history of the physical sciences. If we include the biological sciences as well as the physical sciences, Maxwell’s paper was second only to Darwin’s “Origin of Species”. But the importance of Maxwell’s work was not obvious to his contemporaries. For more than twenty years, his theory of electromagnetism was largely ignored. Physicists found it hard to understand because the equations were complicated. Mathematicians found it hard to understand because Maxwell used physical language to explain it. It was regarded as an obscure speculation without much experimental evidence to support it. The physicist Michael Pupin in his autobiography “From Immigrant to Inventor” describes how he travelled from America to Europe in 1883 in search of somebody who understood Maxwell. He set out to learn the Maxwell theory like a knight in quest of the Holy Grail.</em></span></p>
Maxwell’s Equations in the elegant form found on college student tee shirts and physics classroom posters were not the the way Maxwell wrote them down in 1865. He did not have the benefit of the power or the simplicity of vector calculus. And the idea of fields as environments was then brand new and hard to grasp. But of greater interest to me, beyond the significance and power of symbol systems which have been well known, was Dyson’s recognition that for many, maybe most new ideas, just the process of writing them down for someone to read in paper or book form is not enough. We have to be taught. We have to learn them. Dyson continues.
<p style=”padding-left: 30px;”><span style=”font-size: 14px;”><em>Pupin went first to Cambridge and enrolled as a student, hoping to learn the theory from Maxwell himself. He did not know that Maxwell had died four years earlier. After learning that Maxwell was dead, he stayed on in Cambridge and was assigned to a college tutor. But his tutor knew less about the Maxwell theory than he did, and was only interested in training him to solve mathematical tripos problems. He was amazed to discover, as he says, “how few were the physicists who had caught the meaning of the theory, even twenty years after it was stated by Maxwell in 1865”. Finally he escaped from Cambridge to Berlin and enrolled as a student with Hermann von Helmholtz. Helmholtz understood the theory and taught Pupin what he knew. Pupin returned to New York, became a professor at Columbia University, and taught the successive generations of students who subsequently spread the gospel of Maxwell all over America.</em></span></p>

I highly recommend you read the rest of Dyson’s paper, but for now, I want to consider the question it has prompted. As books have become more interactive, as textbooks become linked to fancy interactive websites, as courses become MOOCs wrested from the tyranny of a 15 week calendar the physical classroom and the format of live teacher; we now see both methods of education in a wide variety of shapes and sizes. So, today, “What is the difference?” “Are we trying to write and publish (perhaps self-publish) a book, or are we trying to teach an online course?”

For me, these questions are not philosophical; they are real. I am in the process of putting together a book/course on the future of education. Since it is about education in this new digital age, the form and format are just as important as the ideas. So I ask your help.

As we learn from the Maxwell’s Equations story, courses help people digest and learn new ideas that simply reading them in a traditional paper or book form does not. The ideas in my vision of the future of education are radical and no doubt in need of something that looks more like a course, but certainly not a 20th century course and even less like a 19th century book. “So what does it look like, I wonder?” “What does the merger of books and courses make?”

The Future of Math Education

Change in the practice of mathematics forces re-examination of mathematics education. Not just computers, but also new applications and new theories have significantly expanded the role of mathematics in science, business, and technology. Students who will live and work using computers as a routine tool need to learn a different mathematics than their ancestors. Standard school practice, rooted in traditions that are several centuries old, simply cannot prepare students adequately for the mathematical needs of the 21st century.

Shortcomings in the present record of mathematical education also provide strong forces for change. Indeed, because new developments build on fundamental principles, it is plausible, as many observers often suggest, that one should focus first on restoring strength to time-honored fundamentals before embarking on reforms based on changes in the contemporary practice of mathematics. Public support for strong basic curricula reinforces the wisdom of the past—that traditional school mathematics, if carefully taught and well learned, provides sound preparation both for the world of work and for advanced study in mathematically based fields.

The key issue for mathematics education is not whether to teach fundamentals but which fundamentals to teach and how to teach them. Changes in the practice of mathematics do alter the balance of priorities among the many topics that are important for numeracy. Changes in society, in technology, in schools—among others—will have great impact on what will be possible in school mathematics in the next century. All of these changes will affect the fundamentals of school mathematics.

Lynn Arthur Steen

ASCD Mathematics Curriculum Handbook 1998

 

20:1

Even knowledgeable math educators have a hard time defining the most important topics in the Common Core Math Standards and its offspring. The “Standards” is a complex and demanding document that its authors try to organize, link, and sequence by writing the progressions, and its proponents amplify in trainings and workshops. This complexity makes it hard for all but a few teachers to order and focus lessons on the critical concepts. They thus follow their textbook’s scope and sequence believing their authors have that special knowledge. And they march their students from page to page of equal weight, homogenizing and leveling the math concepts. Some will use public collections of test questions to enhance student success rates which only serves to  increase focus on those topics that are easy to multiple-choice test. But in the end our students come away with the feeling and understanding that mathematics is the study of lots of small separate ideas and procedures that have little if any coherence.

Fortunately, technology offers us an easy way to find out what mathematical ideas the authors of the Common Core think our kids need to learn. Load its pdf or your state’s framework and use find (Control F) to count the number of times a word or concept is mentioned. I did this for the word fraction(s) and the number I get is 200 or so. Fraction(s) are mentioned in the 93 pages of the Common Core 200 times. It is obviously “important.”

I also counted the number of times ratio(s) is mentioned — 12 times! The authors of the Common Core are telling us something. And we can ask are they right? Should our students spend 20 times as much time on fractions as they do on ratios? Are fractions 20x more important?

Ratios are central to our lives and essential to our quantitative reasoning. Broadly speaking they are the division of two quantities. They take a wide variety of forms. Percents are ratios, rates like miles-per-hour or interest are ratios, averages including batting average in baseball are ratios as are statistical means and standard deviations, unit conversions like miles to kilometers or Euros to Dollars are ratios, most economic indicators even price per share are ratios, as are most significant scientific and mathematical quantities like mass and pi ratios. It is hard to think of any useful math that today that does not involve ratios.

In our schools ratio and proportions (proportions are ratios too of course) are the main new topics for 6th and 7th graders. They have to learn to solve a raft of traditional problems around motion, work, mixture, interest, and conversion. These are generally taught in silos as separate forms and algorithms that students are supposed to understand and master. This is just the beginning, because slopes are ratios which makes linear equations and linear functions fundamentally the study of ratios as is the whole of trigonometry. Calculus, too, is the study of ratios, for what is the derivative but a ratio (dy/dx).

Most of our work and life that uses quantitative reasoning revolves around ratios and remarkably little involves fractions unless you follow recipes at a bake shop or are a carpenter in the U.S. Fractions provided shortcuts to calculation in the analog age, but now in our digital age with machines doing the calculating, they are obsolete. And since they are numbers derived from ratios (rational numbers) isn’t it about time that we reverse that 20:1 ratio and prepare our kids for their future and not our past.

If you were to define a mathematics curriculum for the 21st century what concepts would you focus on — fractions or ratios?