Category: Blog

The Math Guys

Our good friend Larry Myatt, one of the great thinkers and leaders on the future of education, recently sent out a New Years greeting that featured What if Math. We are partnering with Larry on a number of exciting projects in 2020 that we will keep you informed of. Meanwhile, I want to introduce you to Larry via his post and our new nickname. We, too, hope 2020 will be a good year for us all and for helping our kids learn math.

ERC: The Future of Math Education: What If?
“Happy New Year 2020. We at ERC are excited to share news about a new partnership for our Innovation and Redesign Network. This one with What If Math, folks with big ideas, with huge potential to change the way we look at math. Read on!

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?

Art

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

Baseball and Math

If you are a Bostonian by address, birth, or just a connection, you can’t help but be full of pride this morning for your baseball team. The Red Sox were amazing, keeping us up late at night and giving us so much to cheer about in a time otherwise to often dreary. And if you are a math teacher you cannot but wonder why so many of our kids find math irrelevant, uninteresting, and difficult when so much of what they heard and saw as we watched those World Series Games was math, math that helped us understand the game, math that we found fascinating, math that we understood. It is truly amazing how much math there is in our national pastime.

There are numbers and shapes everywhere from the backs of player uniforms to counting of balls, strikes, and outs, numbers for the innings and even numbers for the game, the series, the postseason, and the regular season. There are addition totals shown on the scoreboard for hits and runs along with the count for each inning. There is the shape of the diamond, measurements to the bases, the mound, and the various walls, the square bases and pentagonal plate. This is only the beginning because baseball is all about statistics. Batting averages and earn run averages, hits off right handers and hits off left handlers, even stats about which team will win the series after being down by 2 games. Pitchers throw balls that curve in left or right, go straight and rise or go straight and fall. The speed of the ball is posted for each pitch along with its position in the rectangular strike zone..

The game is full of algebra and not just arithmetic, geometry, or simple statistics. The amazing TV graphics shows the parabolic path of the ball when a player hits a homerun where we think quadratically as we wonder about height vs. distance, or time to land. It is about how long it takes for a ball to be thrown across the infield compared to how fast a runner can get to first base. It is about where on the bat the ball hit and the shape of the swing for maximum momentum transfer. It is today about building models for managing teams, salaries, salary caps, offence vs. defence, the number of starting pitchers you must acquire vs. the number of relievers and even the specialized relievers, the closers. And it is about the wealth of statistics and statistical analysis rivaling the stock market in richness and complexity.

I have only begun to scratch the surface of math and our national pastime. For as I watch and listen to the games, I hear and see math everywhere. How could it then be that our school math is so out of sync with this sport math we learn and enjoy so much? That is the question I leave you with today as I cheer Go Sox as loud as I can and think about how much mathematics helped me enjoy the game.

Here is a sophisticated baseball What if Math Lab you may want to try: Hit Streak.