Logistics Curve #Metaphor #maths #mathematics #nerds #geek

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This is the third in a series of posts on mathematical metaphors. Don’t let the topic scare you. We all use these metaphors, most of us use them every day.

This is a graph of a logistics function. It is not a common function and is rarely discussed. However, it is a good one to learn as it is a much more accurate metaphor.

Just as the exponential takes a broader view than the linear, the logistics curve is broader than the exponential. While small sections of the exponential look like linear graphs, small sections of the logistics curve look like exponentials.

The metaphor recognizes that while things can growth quickly and change rapidly, this condition can not continue indefinitely. Eventually, growth slows down.

For example, Facebook increased its total users exponentially until it ran out of people. When virtually everyone with a phone or a computer was on Facebook, it couldn’t grow anymore. This is the truth of the logistics curve metaphor.

The logistics curve recognizes that things do not expand forever. In politics and economics, this metaphor is expressed as diminishing returns; what work well yesterday, barely works at all today.

This metaphor provides balance. It discourages the winners and encourages the losers.

Reality often matches this metaphor even though most people ignore this model.

The interesting thing is that all three models/metaphors are what mathematicians call: monotonic-they always increase. They never turn around. linear goes up, exponential goes up really fast, and the logistics curve ultimately goes up really slow.

While this arcane curve is a good metaphor, it is nor the best.


Exponential #Metaphor #maths #mathematics #nerds #geek

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This is the second in a series of posts on mathematical metaphors. Don’t let the topic scare you. We all use these metaphors, most of us use them every day.

This is a graph of an exponential function. It is the second most common metaphor, and almost as often wrong as the first.

This is the graph of compound interest. While the previous linear graph is often misapplied to daily life, relationships, jobs, and happiness, the exponential graph is usually misapplied in business and science.

Scientists take a broader view. Even though a small section of an exponential curve looks linear, the whole picture is radically different. An exponential is accurate over a longer period than the linear metaphor, but still misleading.

The population of earth is an exponential. The growth of a successful start-up company is also exponential. In fact, this is the definition of success for most companies. For years the capability of computers matched this metaphor. (See Moore’s Law).

Some people see the world as exponentials. They expect each success to be followed by two more and then four more, as small successes give birth to large successes.

In politics, this metaphor breeds extreme responses to elections, with the both the winners and the losers expecting dramatic changes.

Reality has never delivered on this model of exponential growth for the long term, yet people continue to hold onto this idea.

Stay tuned. There are better metaphors.

Linear #Metaphor #maths #mathematics #nerds #geek

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This is the first in a series of posts on mathematical metaphors. Don’t let the topic scare you. We all use these metaphors, most of us use them every day.

This is a graph of a linear function. It is the most common metaphor, and undoubtedly the most often wrong.

The linear graph says nothing is going to change. This is a graph of a car going at a constant speed…forever. The is total income from the job that never changes. The is the total happiness of a relationship that never changes.

People use this metaphor when their candidate wins an election and they expect everything to go their way from now forward. Alternately when something does not go their way, they foresee constant calamity going forward. This is the metaphor of click bait: “You won’t believe what happened, and is going to continue to happen forever more.”

Reality has never delivered on this model of consistency, yet people continue to hold to this idea.

Stay tuned. There are better metaphors.

A Day in a High School Math Class #STEM #Education

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I am a California Credentialed Math Teacher and substitute teacher. Today I had the privilege of teaching math at the local high school, something I enjoy very much in my retirement.

Today was different because I had just finished reading The Confidence Code, The Science and Art of Self-Assurance – What Women Should Know.

These two events were synergistic.

First, I was encouraged to find equal numbers of girls and boys raising their hands to my question, “Who plans use math beyond high school? Who intends to study Science, Technology, Engineering, or Mathematics?”

It hasn’t always been like this, and I took this as an indication that efforts to interest girls in STEM fields were working. In addition to girls interested in veterinary and pediatric careers, I also found girls in interested in sports medicine and civil engineering.

I also found girls that were interested in math, but getting discouraged. This was my first opportuinity to shared what I’d been reading. Research reported in The Confidence Code has found that boys’ mathematical brain development peaks in high school while the same development for girls peaks later in college. This was encouraging news for the these girls with interest waning due to some struggles.

As I worked one-on-one with individual students, I observed the exactly the behavior that was the subject of The Confidence Code. Many of the girls did not have the confidence to attempt the assignment.

The book reports that girls/women consistently under estimate their competency, while the opposite is true for boys/men. As I explained this to struggling students, we observed that they could actually do the problems that they were avoiding. One of the girls summarized this research with, “So we just psych ourselves out.”

A second piece of research that suggested a very specific action for these girls. There was an experiment where boys tested above the girls on the first trial. On the second trial, the girls were required to answer every question, not leave any blank. In the second condition, everyone all test the same.

Sure enough I found a number of girls who skipped questions. When I sat with them and encouraged them to complete every one, just like in the experiment in the book, they discovered they knew how to work the problems and get the correct answer.

This day was encouraging for two reasons. First more girls are considering STEM fields to the point where they are willing to raise their hand and declare this position. Second the research reported in this book played out just as predicted. This book shows the way to promote success for the increased number of girls now considering STEM.

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Image credit: Joe Mabel [GFDL ( or CC-BY-SA-3.0 (], via Wikimedia Commons

Nobel Prize Winner, Mathematician, John Nash dead

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Riding in a New Jersey taxicab, John Nash and his wife were killed in a car accident. The world already had too few mathematicians. Sad.