Category: 5 As an experimental science

Math is Hard

In a magazine published for college trustees, a recent short article captured the latest statistics from the ACT and SAT tests. The downward trend was notable especially in math. For example, “Forty-nine percent of the class of 2018 that took the SAT (2 million students) showed a strong chance of getting at least a C grade in a college-level math test, much lower than the 70 percent who reached the same benchmark for reading and writing, according to the Washington Post.” Math is preventing our kids from getting a college degree.

But what really broke my heart was this number: “The share of ACT test takers who showed readiness for college math fell to 40 percent–the lowest level in 14 years.” They go on “‘The negative trend in math readiness is a red flag for our country, given the growing importance of math and science skills in the increasingly tech-driven U.S. and global job market,’ said Marten Roorda, ACT chief executive officer. ‘It is vital that we turn this trend around for the next generation and make sure students are learning the math skills they need for success in college and career.’”

This is a great ad for What if Math, where math for the digital age is not hard.

Art

A Very Good Year

I feel most fortunate when I have a year I get to work on a new great idea in it. This past year has thus been one of good fortune. Some great ideas can appear huge from the start, covering wide swaths of life, and some, at first, can seem small, almost insignificant initially, but on reflection turn out to be consequential and central. This year the great idea was the latter, an idea that appeared tiny and obvious at first glance, but that grows and grows, becoming more valuable with each passing day. Some great ideas seem to be wonderful creative inventions, unique and very, very different, while others are quiet, seemingly so common that we easily miss them. This year’s great idea was again the latter. I would never have expected it to grab hold of our minds and continue to pull us in new directions. And some great ideas seem to spring whole formed, exploding like the Big Bang connecting all sorts of wonderful things together. While others, like this one, play hopscotch in our minds, jumping to and fro with little recognizable pattern to see at first, but eventually linking all sorts of ideas together.

This year’s great idea germinated late last spring as Peter and I were working on what we came to call Classic Story Problems, in particular, the familiar motion, work, and mixture problems found in middle school and high school math textbooks. These problems are painful, that is the only way to describe them, painful to learn and painful to teach. You know the type, “George leaves New York and Martha leave Washington at the same time going at different speeds, where or when will they meet?” Students fight hard to figure them out and many, I would argue most, finally succumb and memorize a formula. Then, of course, when the wording changes, even by a small amount, they are lost again.

In the process of building simple tables in Excel with time in hours in the first (input) column and distance in miles (output) connected by the rule that multiplies time by speed to get distance, we realized that we did not need to fix the input values. Originally, as I was starting to develop spreadsheet lessons, I would let Excel create my input column by putting the first and second input values into the column and then dragging the + to create the column of values. As we got more sophisticated we began to use a rule to create the column, add 1 to the previous number. Well it finally occurred to us that it would be very nice if we could enable students to easily change the input values. Why not make a parameter table with an initial value for the input and an incremental value, and build the input table using a formula and those values? This way we could easily change the start value and the increment value.

Seems such a simple change. But what power. We could enable students to easily solve story problems using a table with discrete values by letting them choose the level of accuracy they needed, developing an understanding of accuracy along with problem solving ability. We could enable them to easily change the domain of any function they choose to graph and quickly and easily control their graphs. We could enable them to zoom in on a particular aspect of a function to study it, or zoom out to picture its form. We could enable them to easily ask “What if…” in a myriad of new ways.

This tiny idea, a parameter control table that enables students to change constants as well as variables has great power. Make the increment smaller and smaller on a quadratic function and you can see that segment of the graph becoming straighter and straighter. Change the initial value and you can watch that straight line change slope as it moves around the parabola. It is almost magical. We don’t need to try to get students to understand limits or secants to picture derivatives, or find a common way to solve story problems. And we are giving them the tools to be flexible problem solving thinkers and explorers.

So as Frank Sinatra sang, “It was a very good year.”

Motion Problems

Opportunity for Creativity

I just looked at a wonderful short video by Sir Ken Robinson on creativity at https://youtu.be/63NTB7oObtw in which he describes creativity as a process that produces something original that has value.

At What if Math we seek to make learning a creative experience, a process that enables every student to produce something original that has value. Isn’t that the essence of what learning is or at least should be. Can all learning be a creative experience? Can it be problem-based with interesting problems that can appeal to a wide variety of students? Can it provide powerful tools for students to experiment with and to stretch their imaginations enabling them to produce something original and of value? We believe most, if not all, of it can!

The spreadsheet math problems and experiments in What if Math are designed to enable every student to be creative and to experience creative learning. We hope you will encourage your students to learn and use the wonderful power of spreadsheets to be a creative problem solver. And we hope you will help us create new Labs by sending us problems to develop or Labs you would like us to publish. I look forward to hearing from you.

Art

Another Sunday Ritual Soon Gone

When I was a kid, Sundays in the summer were car washing days. The stores were closed. The roads were generally quiet. And we took out the hose and the pail, filled them with water and dishwashing soap then rubbed, scrubbed, and waxed the family car…or later our own car…beautiful again. Sundays have already substantially changed and now in an article a recent Economist magazine, one of the last of the great suburban Sunday rituals will soon be going away too. For a scientist has found a pattern made by an obscure but available laser that sheds water and dirt with it. One day in the not too distant future we should be able to buy a car that not only no longer needs waxing but will never get dirty and need washing. Perhaps, about the same time, we will no longer even own cars but instead will ask for one on our cell phone, and an Uber Self-Driving car will pull up to our house to pick us up and drop us off wherever we want to go.

Crazy ideas, great changes. Yes, this is just a tiny example of the new world our students need to be preparing for. We are all linear thinkers. We think the world will continue in much the same way it has been going, with changes that take place slowly and methodically, changes that we can get used to, changes we can plan and prepare for. But change is not linear at all. It is, like the word used by the wonderful writer and biologist Stephen Jay Gould to describe evolution, “punctuated”. Sometimes change accelerates quickly and sometimes it moves at a constant speed. This has always been the nature of change and it is evermore so true today. For things that shed water don’t get dirty and don’t need cleaning — cars, windows, even clothes. And cars that don’t need drivers don’t need parking spaces on streets, driveways, or shopping centers. Our world could, and likely will, change dramatically.

How do we prepare children for work and life in this future world? What should they learn in their school years to make them ready for a lifetime punctuated with change they cannot predict? We cannot base it on the Sunday rituals of our past like counting the cash my dad brought home from his dental practice so that he could put it into the bank on Monday morning. We cannot base it on the paper and pencil calculating rituals we spent innumerable hours on, the paper algorithms that define most of the math our students practice. We cannot base it on “What is____?” habits of thought of the past, when the answer can almost always be easily Googled. We must base it on “What if…” thinking, functional thinking, the basis of science, technology, engineering, and math, the heart of business planning and quantitative reasoning, the question the future depends on.

Sunday rituals will come and go. Technology will sometimes change rapidly and sometimes slowly. But we can prepare our children for their future by making their education about “What if…”, their practice and mastery not off paper-based algorithms but off open-ended problem solving, their focus not on facts in today’s data-rich world but on thinking, their vision directed not on finding the right answer but on seeing outside the box.

Tradition, Tradition

As part of the process of designing and developing new Labs, I visit math content sites all the time to help me think about the kinds of questions to ask and the way to explain or represent a concept. I am constantly struck by how talkative these sites are. As teachers, words are our currency, and with few exceptions they are the main way we have always communicated skills and ideas. We come from a very long “stand and deliver” tradition. We seek to replicate Socrates talking to his disciples. When that oral tradition was turned into a printed one, teachers used words, even more words, to communicate ideas. Surely, in some cases, we draw pictures, knowing a picture is worth a thousand words, particularly in mathematics. But we rarely let images stand alone, but embed them in a sea of words, for words have remained our currency and our tradition for 2500 years.

Today, as we turn into the digital age using screens instead of dead trees, we continue to find it so very difficult to get past our tradition. We make videos, draw and animate images on screens, but still we fill them with words. Even when we create content with dynamic, interactive images, we still embed them in a sea of words to either be read or listened to. We, it seems, cannot leave our long tradition of making words our learning currency. Even when our visionaries preach teaching in the tradition of the great Socrates by asking questions, having conversations, seeking roots of concepts, we continue to apply his words based pedagogy to build 21st century skills.

We have yet to learn the lessons of this new digital medium, the lessons of PowerPoint slide shows, Twitter and Facebook, emails and especially messaging. We have not applied the “less is more” use of words to digital learning. It is not easy to make lessons with just few words that do not have to tell, show, or direct. It is not easy to ask simple questions that suggest. It is not easy to picture concepts in visual representations as tables or graphs or animations. It is not easy to change tradition. But just as in Fiddler, we must!

“Just try it on!”

Spanglish is one of those movies that grows on you. A coming to America story filled with themes that move us: a dedicated and resourceful woman, a dysfunctional but caring family, a highly successful artist, and of course love. It has many scenes that touch us deeply. One of those, highest on my list, is when Flor, who has just started as a housekeeper at the Clasky home, comes back to her own house after witnessing the mother embarrassing her daughter over her weight by buying her clothes a size too small. Flor, who had never learned English, asks, no she demands, that her daughter teach her how to say, “Just try it on!” They repeat it in synchrony over and over again. Flor arrives at the Clasky house before dawn to let out the new clothes, then she wakes Bernice up and holding up the clothes, she speaks English for the first time. “Just try it on!” “Just try it on!” she demands again and again until Bernice finally succumbs and the smile returns to her face.

I think of that poignant scene when I contemplate students and teachers in math classes today, for in conversations with teachers, parents, administrators, and yes students, I hear story after story about classrooms filled with kids who have given up on their ability to learn math, are just plain bored, or who see no reason to learn the math they are being taught. Teachers, who are themselves bored with a curriculum so heavily structured and predetermined it leaves no opportunity for creativity or even fun, try to pretend they are not. It is clear that teachers, students, parents, and administrators are dealing with a subject that no longer fits, a subject which has shrunken beyond recognition, a subject no longer relevant, no longer meaningful. Though, it claims to be conceptual, to enable students to learn to think and to solve problems, in reality it is mechanical in an age where machines have taken over most of those functions. It is too small for our digital age students and teachers who are wanting more, more relevance, more creativity, more fun, more learning. And it is too small for our classrooms where teachers want more discretion and more opportunities to engage.

In What if Math we have done more than just let out the old curriculum. We have started from scratch to build a digital age curriculum. Did we get it right? Do your students enjoy it more, learn more, feel better about their math ability? Are you having more fun? There is only one way to know, so I ask you, “What have you got to lose?”, and as Flor coaxed Bernice, “Just try it on!”, “Just try it on!”

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

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