Author: Art Bardige

I am a digital learning pioneer who believes that technology can play a great role in enabling every child to learn efficiently, effectively, and economically. What if Math is my latest work and the most exciting I have ever been involved with. I hope you will give it a try.

NCTM Nashville October 4, 2019

Exploring Mathematical Concepts with Dynamically Linked Multiple Representations Using Spreadsheets
Fri, 10/4: 11:00 AM – 12:00 PM
60 Minutes
Music City Center
Room: 207 AB
How can we build multiple representations into teaching and learning activities? This can be a challenging task, but technology provides the means to do this effectively and efficiently. Join us in for an engaging interactive session to experience how a wide range of math topics can be dynamically represented and linked using spreadsheets.

Black Hole

This picture made the front page of the New York Times this morning. It is not very often that a science experiment makes the headlines in a great newspaper, particularly above the fold. It is a thrilling discovery. “The First Image of a Black Hole.”

The concept of black holes, hundred years old, came out of Albert Einstein’s General Theory of Relativity equations. Gravity, the primary force in the universe depends upon mass and distance. If enough matter is squeezed together tightly enough then any light emitted by the object will not be able to escape. All light and all matter will stay in this black hole.

This black hole in the center of the galaxy M87 was captured by a unique array of telescopes two years ago. Most galaxies have black holes at their center. This one in M87 is larger than the black hole in our own galaxy, The Milky Way. The black hole in M87 is nearly the diameter of our solar system and has is several billion times more massive than our sun.

  1. To begin to get some idea about the size of this black hole, I suggest you go to the Solar System Lab and explore the size of this black hole and the size of our sun.
  2. If the black hole is the size of our solar system, can you build a model that will calculate the number of suns it would take to make it.
    Make a new table of black hole data like the Solar System table.

  3. How far away are we from the M87 black hole compared with our distance to the sun?


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.

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