Inside of a pipe overlooking water

How Mathematicians Think: Abstracting out the Big Ideas

Karoline Pershell
Sep 27, 2013

In two days, I had two conversations with two people over two really neat technological developments . . . which turned out to be the same process!

When a beautiful idea springs up independently in different situations, step back and take note of the greater principles at play, asking if they can be generalized elsewhere.


Conversation 1: Pipes underground may degrade over time (corrosion of metal or degradation of clay, tree roots infiltrating or collapsing the pipe walls). Repairs tend to be expensive as the entire area needs to be excavated and new pipe laid.

A technique (shown here) runs a line through the damaged pipe and pulls the makings of a new pipe through the old pipe. An inflatable bladder is covered with a “pre-concrete” material that—once in place—is activated and sets up around the bladder, which is then removed, leaving a recently paved water (or sewage) highway.

Done in a short time, no need to have extensive teams and equipment, no need to rip open the street, and no need to wait weeks for landscaping to be restored.

Immediately fixed.



Conversation 2: Valves in your heart may degrade over time (high blood pressure, genetic predisposition). Surgery plus recovery is financially and physically taxing as the entire chest cavity needs to be opened and new valves inserted.

A technique (shown here) runs a line through the femoral artery to the damaged valve and pulls the makings of a new valve through the old valve. An expandable trap is pre-stretched with a manufactured valve that—once in place—is activated to spring open, expanding the new valve in the exact location as the old valve.

Done in a short time, no need to have extensive teams and equipment, no need to open the chest cavity, and no need to wait weeks or months for recovery. 

Immediately fixed.

Double wow.


Both of these examples demonstrate how far imaging technology and materials science have come.  Let’s examine the big problem and the solutions.


Problem: The infrastructure necessary for normal functioning is wearing out. Typically the efficiency will slowly diminish and then the system will have a sudden failure.

Helpful factors: The diminished efficiency can be recorded, and intervention can happen well before failure occurs. The infrastructure is already in place and does not need to be created, just repaired.

Solution: Go through the working infrastructure to reach the point of expected failure rather than shut down and open the whole system (which is costly, time consuming and dangerous). Have the replacement apparatus prebuilt and then assembled on sight.


Where else can this idea be used? What cues can you find in a problem that would tell you to try to connect your problem to these methods? Ailing infrastructure? Decrease functioning that precedes failure?

Don’t limit your thinking to what you know we can observe or to the materials at hand. If you can see a solution, there are fields of science just waiting to create the technology and tools you need!

With trained creativity, we can learn to abstract a problem out of the context in which it was created, freeing us to use tools in other disciplines that we might not have considered! For example, just knowing about leaky sewage might not have made you think about the heart.

{And if it did, you are weird.}



** Or “Your heart and the feces under your feet have a lot in common!”



Karoline Pershell

Karoline Pershell is a current AAAS S&T Policy Fellow at the Foreign Service Institute. Prior to her fellowship, she was an Assistant Professor of Mathematics at the University of Tennessee at Martin (UTM). Karoline was invited to teach summer courses at Qingdao University (2012) and just completed a position as a Fulbright Visiting Lecturer at the University of Hyderabad, India (2013).

Raised in Michigan, she transferred to Tennessee in 2000 to pursue bull riding, becoming the first woman to ride bulls in the Ozark Region of the collegiate circuit, and winning Rookie of the Year (2001) in the Professional Women’s Rodeo Association. She received her Ph.D. in 2009 from Rice University. Her interests are: inclusion of disenfranchised groups in higher education; math, logic and communication skills as a means of empowerment; communication between the scientific and non-scientific communities; implementation of green building practices.

When not working, she is renovating her home and spending time with her 1.75 dogs, contemplating how to deal with big data. Follow the transition from academia to policy


This blog does not necessarily reflect the views of AAAS, its Council, Board of Directors, officers, or members. AAAS is not responsible for the accuracy of this material. AAAS has made this material available as a public service, but this does not constitute endorsement by the association.

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