Category: 4 Changed the World

Calculus for 5th Graders

“Calculus is the study of change” is a common description and all too common misunderstanding. For if calculus is the study of change then algebra becomes the “study of symbols?” I believe this confusion between algebra and calculus among math educators makes these subjects, and the choice to study them, a challenge to students. It is the reason some even advocate changing the traditional arithmetic-algebra-calculus sequence into arithmetic-algebra-statistics, believing statistics to be more important and more relevant to the everyday lives of students. I think they are wrong.

If we say instead: “Algebra is the study of change.” thenCalculus is the study of the rate of change.”

The symbol we associate with algebra, “x”, was defined by Descartes in 1637 as the symbol for a variable. Unfortunately, because today x also used to represent an unknown quantity as well as a variable, our students are confused about its meaning and the nature of algebra. For those who understand the language and concepts of mathematics, numbers represent fixed quantities, and variables like x represent changing quantities. Our students who ask, “What is x?” will likely describe math as abstract and early on make the decision that they are a member of the majority who just don’t get math.

Spreadsheets, work with discrete variables. A number or an unknown is a fixed quantity in a single cell. A variable is a row or column of numbers, a table of values. This fundamentally visual and concrete representation of variable makes the difference between an unknown and a variable vanish, and this all too often asked question, go away. Function, symbolized as f(x), often baffles students. A function can be visualized by the second column in a simple table, next to the first one, an “output” generated by a rule that transforms a variable (input) column into an output column.

We can graph the function easily on a spreadsheet. And we can make a table for its parameters so that we can change and adjust the function easily. This format gives students the ability to play with functions and to learn to visualize them and build models using them. For a mathematical model is built out of functions.

The derivative, a special class of functions, rates, ∆f(x)/∆x, could be a third column taking the differences between the function values and dividing by the variable differences. If a student makes ∆x, the interval between the input values, smaller and smaller she will be zooming in on the function and build an intuitive sense of limit and derivative. The derivative is now clearly not the study of change, it is the study of rate of change.

If rate is a concept we expect 5th and 6th graders to understand, then derivatives using discrete variables on spreadsheets is a concept they can understand as well. And they can be asked to solve real problems that ask for rates of change.

Spreadsheets enable all our students to get math, to see variables, functions, and derivatives as concrete easily calculated mathematics. And all can be easily envisioned graphically.

Try it yourself. Build a spreadsheet like this one and experiment by finding the slope of the curve at different points on the curve. Was Jim Kaput right. Can these fundamental and important ideas of algebra and even calculus be learned by 5th graders.

Spreadsheets and the Rule of Four

A little over 20 years ago the Harvard Calculus Consortium sought to remake the calculus curriculum. “We believe that the calculus curriculum needs to be completely re-thought,” began the text by Andrew Gleason and Deborah Hughes Hallett, both of Harvard University. They sought to get “our students to think.” In doing so they proposed “The Rule of Three.” “Our project is based on our belief that these three aspects of calculus—graphical, numerical, analytical—should all be emphasized throughout.” The Rule of Three, today often known today as The Rule of Four with the now addition of verbal, rests at the heart of math education. While the Calculus Consortium’s book may no longer own major market share, it has had a remarkable influence on all Calculus textbooks and indeed on all math textbooks in both K-12 and college. It is a widely shared belief that such multiple-linked representations must be central to 21st century pedagogy. It is clear that students learn in different ways. It is certain that they need to see mathematics from different perspectives.

Spreadsheets are Rule of Four platforms. They are function machines which naturally represent mathematics graphically,
numerically, analytically, and verbally. They show a function as a graph, as a table, as a formula, and we can describe them with text and visuals. They did not start out that way. The first spreadsheet, VisiCalc invented by Bob Frankston and Dan Bricklin was designed to be a visual calculator to automate the accountants’ worksheets. Three years after VisiCalc’s debut in 1979, Mitch Kapor added graphs and tables to create Lotus 123 which brought the IBM PC into every business. And Excel from Microsoft came out for the new Macintosh 2 years later not only simplifying the interface but adding beautiful texts and visuals to spreadsheets. Today, the mature spreadsheet technology is the standard quantitative tool for business worldwide. It is not only available on every major platform, but its format and design are the basis for displaying and interacting with quantity on the Web.

In a spreadsheet we can write a formula, use that formula to create a table of values, and use that table of values to make a wide variety of different graphs and charts. Change the formula and the table and graph changes automatically. Change the table and the graph changes automatically. Spreadsheets are dynamic and highly interactive. They even let you embed variable quantities in text to add units to quantities our dynamic values to verbal descriptions. Once a student builds a model in a spreadsheet, it is naturally a multiple-linked representation that can played with and explored. Spreadsheet models designed with functional thinking as multiple-linked representations are therefore simulations of which students can ask “What if…”

If you use Link Sheets in your classroom, if you believe that every student has a learning style, if you like to have students explore different representations, if you want to get your “students to think” then try using our What if Math spreadsheets or develop your own built on the Rule of Four.

The First Graph

This picture was first published in 1638! It is from Galileo’s great work Two New Sciences, that he smuggled out of his home imprisonment in Florence, when he was 72 years old and effectively blind. Though famous for his telescope and the first images of the surface of the moon, he had not before published his seminal work on motion. It is easy today to gloss over his extraordinary achievements, even being called the “father of science,” and to make him a caricature battling for the Copernican theory. But this view of the renegade, the persistent critic who fathered three illegitimate children, fails to recognize his profound contribution to humanity. Galileo invented the experiment. Before him people “observed” nature. They developed instruments that simulated phenomena like the motion of the planets. But they did not “experiment.” They did not ask “What if…?

An experiment is a process in which we can change not only inputs, but the rules connecting inputs to outputs as well. Galileo’s classic experiment, the motion of objects sliding down inclined planes, enabled him to dilute gravity, to slow down the motion of a falling body, so that he could measure the distance traveled in each time unit. To do this: he built the first accurate way of measuring short periods of time, he constructed an inclined plane so he could ask “What if I change its slope?” He developed the concept of repeated trials to measure, re-measure, and measure again so fundamental to experimental science.

In the fourth chapter, which he called the fourth day, in Two New Sciences (the first new science being the science of proportion) on the science of motion, he showed how projectile motion, the motion of objects shot out of cannon, thrown, or dropped, can be envisioned as the composition of two motions horizontal and vertical. The horizontal motion of a projectile is constant, it goes the same distance in every unit of time. The vertical motion of a projectile, like the motion of any falling body is accelerated, the distance it travels increases as the square of the time. Added together the motions to produce the path of the projectile and that path is a parabola. We would call the graph that Galileo drew a distance/distance graph (both axes are distance measures). At each point in time, we move across and down. This compounding of motions like the compounding of functions in the Parametric Equations Lab enables us to put together two separate functions linked by a parameter (a parametric variable) to model motion.

As you experiment with the Parametric Equations Lab, imagine you are Galileo, experimenting with inclined planes, dreaming of dropping balls from the Leaning Tower, and explaining why, if Copernicus is right that the earth rotates, we don’t feel ourselves moving. And as you experiment replicate has graph, the first ever drawn and imagine how he would have used it to explain the motion of projectiles.

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.


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…”


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