Category: 3 Make Room

The Hawthorne Effect

To make its workers more productive, the Western Electric Company, makers of phones and other parts for the Bell Telephone System, conducted one of the great scientific experiments of all time. The researchers increased the brightness of the lighting at a plant in Hawthorne, Illinois just outside of Chicago, incrementally, while measuring worker output. They alerted the workers to the experiment and they found that as the workplace got brighter the output increased. Then, in a great out-of-the-box experiment, they dimmed the lights incrementally, telling the workers the experiment was still going on. Output continue to grow!

This became known as the Hawthorne Effect.

…the novelty of being research subjects and the increased attention from such could lead to temporary increases in workers’ productivity. Wikipedia “Hawthorne effect”

I got to thinking about the Hawthorne Effect while rereading the Common Core State Standards and thinking about the testing controversy that embroils us. In their website under the Key Shifts in Mathematics menu, they shine a bright light on what they consider to be the core topics:

  • In grades K–2: Concepts, skills, and problem solving related to addition and subtraction
  • In grades 3–5: Concepts, skills, and problem solving related to multiplication and division of whole numbers and fractions
  • In grade 6: Ratios and proportional relationships, and early algebraic expressions and equations
  • In grade 7: Ratios and proportional relationships, and arithmetic of rational numbers
  • In grade 8: Linear algebra and linear functions

And they conclude:

“This focus will help students gain strong foundations, including a solid understanding of concepts, a high degree of procedural skill and fluency, and the ability to apply the math they know to solve problems inside and outside the classroom.”

The Hawthorne Effect caused by these brighter illuminations will no doubt improve student productivity as measured by test results. But as with the Hawthorne plant, once the experiment, and it is an experiment, is over, once the lighting no matter what the brightness level, stays constant, then productivity will return to its “normal” level. For we have not really changed either the product to make it much more integrated nor have we changed the process to make it more productive.

For the past 50 years we have put math education through a series of Hawthorne Effects. This is the reason that the NAEP scores have remained flat lined. If we truly want to substantially improve math education, as we must, then we will have to reinvent both its product and its process and not just shine brighter lights on it.


As I watched a young woman the other day learning to ride her bike, zigzagging down the street, desperately trying to keep her balance, I thought of the Wright brothers. They owned a bicycle shop where they made and taught people to ride the then relatively new form of transportation. They, like their students, had to learn to balance these new contraptions, just as most of our children do today. They took that core skill to their work on flight. For they were not the first to try to solve the problem of mechanical flight, but they were the first to be successful at it.

That success was built on many things, on a sense of design, on an understanding of both the physics and the engineering of lift, torque, control, and material science, on observation of birds, and even on the design and manufacture of the then very new gasoline engine. But if we had to ask, “What singular idea enabled them to be the first flyers?” The answer would have to be balance. They would have taken that from their bicycle business for it is also the secret to riding a bike. The small wing sticking out in front of the main wings had a reverse curvature to balance the lift of the big wings. The wing warping mechanisms enabled them to balance the chaos of wind and air currents and use a tail as a rudder to turn. They balanced weight, size, and structure to make it possible to carry a pilot and a passenger. And they even had to balance success and failure, secrecy and publicity, business and invention.

Education today is off balance, and like the Wrights we too have to make balance not just a priority but our central driving force. The balance between the arts and the sciences, so long a key aspect of our educational system is now gone. The practical arts are no longer part of our school day. The fine arts are for all intents and purposes missing in action. And the design arts, so critical to business today are not found in our schools. The sciences which includes math, for math is the “science of patterns,” have taken over our curriculum. English Language Arts, is not artistic but scientific with word counts, difficulty formulas, and non-fiction governing that subject.

We try. We add an A to STEM and make it STEAM, but we don’t seem to have a clue about how to integrate those subjects. We talk about the importance of the need to bring the arts back, but we are so nervous about those tests that we cannot find the time in the school day to do that. And while private schools flaunt their wonderful studios, their darkrooms, their theaters, and so many of the people demanding charter schools seek to emulate the privates, the measure of a good high school remains its SAT and AP scores.

As we invent an education system for the future and not the past, we, like the Wright brothers must make balance our central design principle. Just as balance enabled the Wrights to create a new form of transportation, so too must balance, by infusing our classrooms with the creativity of the arts, enable our students to fly.

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!


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.


“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.


Collaboration is Cheating?

One of the four C’s, perhaps for many the most important 21st century skill, is considered in our schools, cheating. Students caught talking to each other during exams are either yelled at or disciplined for cheating. Homework is supposed to be an individual activity and students are punished for cheating if their paper looks like another. In English or social studies, if you are caught copying something or someone without attribution then you are plagiarizing and treated as if you have committed a crime, cheating. And if you are doing a project with a group of students, be sure your work and effort are your own not the work of others or you are cheating. We are training students from the earliest school age to work individually, to “do your own work,” to not cheat. We are still teaching our students to be rugged individualists, independent, self-motivated, and self-reliant. We are teaching 19th century skills.

Today, collaboration is one of the four C’s skills because it is seen, in survey after survey of business, to be critical to digital age problem solving. Creative problem solving is considered a group activity today, and business would no more consider isolating individuals in the workplace than taking them off the Internet. Offices and universities are designed to breakdown silos, to have courtyard and corridors, like this design of the new Google headquarters in London, where people can constantly meet, share ideas, and engage in group problem solving. The best employees are considered to be the ones who work well in teams, who are good collaborators.

Yet, we educators act as if collaboration is either a skill we are born with or magically gain when we require it. Despite the importance given to teamwork and collaboration in sports, we still do not consider it a skill we should learn in school, a skill we should practice in school, a skill that is no different from reading or numbersense. Learning to collaborate in school as a central mission certainly requires us to rethink education from the ground up. But even if we are not ready to take on that big task, we can start by making our classrooms meeting places where silos are not just torn down between subjects but torn down between students, where students are supported and encouraged to learn to collaborate.

The Democratization of Knowledge

On this 10th anniversary of the iPhone it is worth remembering that this invention, as world-changing as it was, will not be deemed the most important one of that decade. It will, in the long thrust of history, take second place to an event that truly and profoundly changes the course of human thought. For in this first decade of the second millennium we will be most affected by the democratization* of knowledge. Before this decade, knowledge was rare, a scarce and valuable commodity to be bought, sold, and transferred in schools, workplaces, and books. It was owned by the few and sold to the many. Our schools sold knowledge to students in the form of teacher presentations and textbook descriptions. Our workplaces sold it as apprenticeships, training, and professional development. And our encyclopedias, sold these compilations of all knowledge door-to-door, and libraries were handsomely constructed to give all the opportunity to acquire it in printed form.

During the first decade of our new century, knowledge ceased to be owned by the elite, ceased to be rare, ceased to be precious. It is now owned by all, ubiquitous, and, for all intents-and-purposes, free. Wikipedia, a tiny but highly significant reflection of this revolution, began in 2001. Its rate of growth, as we can see from this fascinating graph, inflated around mid-decade and is even now slowing down as it has filled up with much of the past knowledge. And Wikipedia is but one of billions of posts and sites that today freely contribute knowledge in every sphere of human endeavor. For knowledge continues to expand at, what to most of us seems like an accelerating pace. Knowledge for the first time in human history, and for as far as we can see in the future, is now democratically produced and distributed.

From an educator’s perspective, this revolution profoundly changes our job. We can no longer be paid to carefully dole out the precious knowledge, mainly in text form that we acquired from our sweat and treasure, for it is now freely available on the Web. As painful as this may be to those of us who are or have taught, we will no longer be valued for what we know and show, but instead for how we help our students find what they want or need to know.

Applying this to problem solving demands we change our role from telling to asking, from answering to encouraging, from giving to wondering. Our job is now to encourage our students to use the Web to find things out, to learn how to solve a problem, to search for conceptual understanding, to explore possibilities, and perhaps most of important of all, to choose problems of interest to them to practice and build their skills. While students likely know Khan Academy and YouTube videos, most still need to be invited and yes encouraged to use all the tools available on the Internet to solve problems (in our case to learn to use spreadsheets to solve problems). In the digital age, they and their collaborators need to learn to aggressively find and use the digital opportunities to work on and solve problems. They need to learn to separate good from bad, valuable from worthless, helpful from wasteful. They need to be encouraged to find things out for themselves, to try again, to try harder, to keep trying. And they need to be told to get help from their peers and from experts if they need it. They may, of course, at times need some direction, need us to bend their path a bit, but we will have to remember it is their path and not ours!

*I borrow this term, as it applies to education, from my dear friend Jim Kaput.

Real Feedback vs Artificial Feedback

Math Blaster was the biggest hit educational product in the 1980’s, the first decade of the personal computer age. Flying saucer like objects would vaporize before your eyes when you solved a simple math question. It thus gave you immediate feedback and like pinball, it kept score. My friend Jan Davidson designed and built this first in a lineage of educational products.

Today, most of our digital learning programs pretend to be more sophisticated. They deal with multistep problems and less rote practice. Their feedback is less shoot ‘em up, and their interactions are no longer pure gamey response. Yet they remain wedded to the same kind of “artificial” feedback where the computer knows the answer and students guess it. Students may look like they are doing authentic problem solving, but they expect their computers to check each step they take, and with the same artificial feedback keep them on track to solve the problem. The computer now knows not only the answer, the product, it knows a particular process as well, and it tries to ensure that students get the problem right by following that process.

Unfortunately, in the real world outside of schools, we seldom get artificial feedback to help us solve problems. We must learn to use real feedback to keep us on the right track, to help us solve real problems, especially those requiring creative solutions. We need to learn to build real feedback into spreadsheets to catch selection and rule mistakes, so we learn to build in redundant calculations as checks. We make graphs to give us feedback, to see whether all our numbers fit the pattern. Though spreadsheets may seem intelligent, we have to constantly find ways of assuring that we have not made mistakes in the models we build, so if we are good spreadsheet developers we us headmath to check whether our algorithms make sense.

Students need to be learning to build feedback skills into their problem-solving processes and strategies. Artificial intelligence may be an interesting arena today, but our kids need to learn to create and use real feedback and not rely on any program’s artificial feedback if they are to become good problem solvers.

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.



The words curious and curiosity do not appear in the Mathematics Common Core Standards document, yet they are arguably the most important words in mathematics education. If there is any single habit of mind or critical skill I want our students to learn, don’t you agree, it is to be curious, to question, to experiment, to wonder, to imagine what we will do or become. Isn’t this what it means to reason quantitatively?

Curiosity is certainly the most human of traits. It is the reason we explore, the reason we invent, the reason we question, the reason we love mysteries and games. We are naturally curious. And curiosity is at the heart of mathematics as well. We wonder if particular numbers form a pattern, we want to know whether all triangles that fit in a semi-circle are right, whether all right triangles have two sides that when squared are equal to the square of the longest side. We wonder if our business will make money or our budget will enable us to buy that new computer. We are curious big and small, is the universe infinite or finite, is the chance of winning the lottery worth the price of the ticket, is global warming really that bad.

At What if Math we are all about curiosity. How many of the products in a 12 by 12 times table are odd numbers? If we choose any whole number and divide it by 2 if it is even and multiply it by 3 and add 1 of it is odd will repeating that pattern always make a sequence, 4, 2, 1…? What if we make a table using 1 simple rule, add the two cells in the row directly above it? What does the graph of a quadratic equation do if I change the b term? Was Ted Williams or Joe DiMaggio a greater hitter in the 1941 season? Was Napoleon right about using the Great Pyramid to build a wall around France or Moore right about the exponential rate of growth of microprocessors? What does absolute value do to the graphs of polynomials? Is the rate of change of C02 increasing or decreasing? Should I lease or buy that new car?

If curiosity is so central to our lives, then I am curious to know why it doesn’t appear in our math standards or in all too many of our math classrooms. I am curious to know whether students can care about what they are learning in their math classes if they are not curious about the problems we give them or the concepts they are supposed to learn. I am curious to know whether learning math can be fun for kids if they are not curious about it. I am curious to know…


The Hardest Question

What is the hardest question a teacher has to answer?

As teachers, especially math teachers, we face this most painful question all too often, rarely do we have a good answer to it, and even more rarely does our answer enlighten students. The question is less a query and more a bleating call for relevance. For students, it is one more reason to ignore or at best inattend to a teacher’s presentation. For teachers, it reaches into the deepest sense of who we are and what our job is. Our inability to answer it directly and cogently can feel like a failure in the traditional analog classroom where our primary role is threefold, present concepts and skills, develop examples as prototypes, and motivate students to attend and learn. Our inability to give meaningful answers to the hardest question destroys their faith and our ability to motivate.

I always found the effort to motivate learning to be the most taxing part of my job as a teacher. Learning is work, pure and simple, and to get students to learn we have to get them to work, motivate them by placing what they have to learn into context, making it part of an interesting story, providing a strong rationale, finding a reason for students to care, or as the last resort bribing students with grades.

In our digital age the Web is the resource center, a library for concepts and skills. Whether the result of a Google search, a YouTube video, or Facebook connection, students today can find information, concepts, and even demonstrations of the skills they are to learn. The traditional roles we long assigned to teachers as presenter and prototyper are now generally obsolete. (Technology has even replaced most of the traditional teacher evaluation role with computerized high-stakes tests.) The motivator role remains, which brings me back to the hardest question teachers face all the time.

“Why do I have to learn this?”

Difficult, if not impossible to answer in the traditional analog classroom, it can be even more painful in a digital one that follows the traditional form, because in this time of accelerating change, amazing tools, the Web as an infinite library, and cell phones as communicators, our usual answers are all too often irrelevant!, Therefore, in the digital age, the student who has to answer the hard question. The student has to find his or her own meaning, we cannot give it to them. And to do that the student obviously has to have interesting assignments and choice.

The only way I know to give students interesting choices is to make our assignments project-based! So when you next get that hard question, think digital and think Project-Based-Learning.

The Problem with MOOCs

When MOOCs were the rage in higher education, I asked my friend David Kaiser, a physicist and professor of the history of science at MIT, when he was going to do a MOOC. Dave has won teaching awards at MIT and writes brilliant books on the history of physics. Who better to do a MOOC or two bringing his wonderful style of teaching and presentation of important physical ideas to more people. But he was not at all interested, and as far as I can tell several years later has not done any.

“Why” I asked. “Because you can’t change them.” he replied. As he explained, one of the most wonderful aspects of teaching a course year after year for a great teacher is the opportunity, indeed the necessity, to change and adapt the courses in general and the presentations in particular. His reaction brought back a vivid memory of my first couple of years of teaching high school physics. I usually carefully prepared my lectures which were the standard fare for most of my classes. Occasionally too busy, too tired, or too lazy to develop a new one, I would grab my lecture notes from the previous year which I thought pretty good. The class usually started all right, but I soon got into trouble. The coherence was gone, the presentation no longer seemed to make sense to me. I don’t know if my students realized that I was stumbling, they were too busy taking notes, but I did. So, I would stop lecturing, told my class what I had done, apologized, would come back the next day with a fresh lecture and gave them time to work on their assignments. One of the things that makes teaching such a great job is the year-to-year, day-to-day, and even student to student opportunity for improvement, for growth, for learning. This has not been true of curriculum.

MOOCs like textbooks are expensive to produce. They are linear, moving from topic to topic in a standard form, a continuous line of lesson following lesson. They are thus difficult, often impossible, to update or change. Once created, except for minor revisions they are for all practical purposes, fixed. Yet, the world is constantly changing, and even more importantly students are constantly changing. A fixed curriculum or presentation cannot work. It will no longer work to expect textbooks to have a 7 year lifespan. Nor will MOOCs, made once and used again and again, work either. The analog continuous linear sequence of lessons that represent a course is no longer functional in the digital world.

The digital world is a discrete world. It needs education to be flexible, easy to change, constantly renewing, and growing. The metaphor for the analog age and the MOOC is the book, done once and then published. The metaphor for digital age educational content is the newspaper, renewed and reimagined everyday. One is fixed, unchanging, the other constantly refreshed. One is designed to be the same for all students, the other can be different to suite the needs and interests of every individual student. One is the education of the past, the other is the education of the future.

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.


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?