Mathematics is linear, black and white, determined. Right? Or is it limitless, mysterious, even majestic? While the answer to that question may depend on who you ask, you may find that the answer is “both”.

Mathematics is a rigid set of rules that governs the fabric of all that exists (as far as we know), a non-contradictory structure that establishes order and enables flourishing. It is also an enigmatic field, filled with undiscovered complexity, perplexity, and beauty. Like all of the hard sciences, mathematics is “discovered" rather than invented. Its outputs, theorems rather than theories, a distinction that separates opinion, conjecture, and uncertainty from truths that can be proven.

And yet harnessing the complex power of mathematics at its limits requires a great deal of creativity. Mathematician and theoretical physicist Brian Greene put it this way: “You are, in a sense, straightjacketed by the rules of the mathematics. But within that constrained environment, it’s up to you what to do with the symbols.”[1]

Albert Einstein famously developed his most extraordinary theories out of imaginative thought experiments. The concept of special relativity was born out of the idea of lightening striking a moving train appearing different to someone inside the train versus an unmoving external observer.[2] Einstein understood gravity in a remarkably insightful new way after imagining a person falling from a building, cascading toward the ground at the same rate as an object weighing much less than he.[3] (Note that although these are theories of physics, physics and math are closely related. One could argue that physics puts skin and flesh on the bones of mathematics.)

The use of imagination in mathematics is not only a gateway to beautiful, undiscovered theories, reserved for the most advanced minds. An imaginative student at the youngest age can harness that ability to undertake a very closely related engagement: extrapolation.

#### Extrapolation Everywhere

Extrapolation involves applying a principle beyond an initial example. To extrapolate means to draw conclusions that may not have been initially provided, but that are invariably deducible. If you are given a day’s sales for a small shop and are told that sales follow a predictable pattern, you can extrapolate a week’s, month’s, or year’s sales. If you are shown the back of a door with a deadbolt switch, you will likely extrapolate that there is a front to that door with a slot for a key. We may not realize just how often we are extrapolating, and how crucial that skill really is.

Likewise, students may not hear the *word* “extrapolation” until high school, perhaps when they are being told to imagine that a line segment goes on forever and the graph in front of them is a mere sample. However, the *use* of extrapolation is engaged much earlier, especially under visually-focused math instruction such as Singapore Math.

#### Extrapolation Instruction

At Mentorhood, we engage students with word problems and visual questions, largely for the express purpose of engaging their extrapolation skills. Problems of this nature allow us instructors to view a concept from multiple angles that show the same idea to be true.

Extrapolation enters the equation from the very first mathematical lessons: counting, and addition and subtraction. Once a small child gets the hang of counting to, say, thirty, they are not too far off from the remarkable insight that this counting pattern goes on forever: up the tens by one, and repeat the ones column from zero to nine again. They may get stuck at the odd transitionary number that they don’t know yet, but they understand that by repeating the pattern they’ve learned, they can reach anywhere on the infinite number line.

Number bonds serve as a perfect environment for the first arithmetic extrapolation to be introduced. Starting with addition, you can teach a child that the two small numbers add up to the big number. What happens when you supply the big number, but take one of the little numbers away? We can teach that the logical conclusion is that the missing little number must be the difference between the other two. This is the first example that many students face of looking at the same mathematical truth from multiple angles.

Moving into higher grades, students may be taught basic geometry rules and asked, like with the number bonds, to fill in different kinds of blanks. Understanding that volume is length times width times height enables a student to find the volume if only given the base area and the height, or to find a surface area given the volume and the knowledge that the object is a perfect cube.

This multifaceted approach to concept reinforcement trains students to think in three dimensions and to practice if-this-then-that logic. This type of thinking is crucial for true understanding of a mathematical concept. And it is essential for solving elaborate problems where not all of the information is given, or where there is ambiguity in the premise of the question itself. Beyond pure mathematics, the ability to extrapolate based on logical reasoning is essential to careers in law, computer science, engineering, investigative journalism, and countless other areas.

#### On and On

When a thinker becomes comfortable with provable ideas beyond what can be immediately seen, they can venture into the territory of potentially provable ideas beyond what can yet be demonstrated. This is the imaginary space that Albert Einstein so famously occupied. Those at the cusp of quantum mechanics or energy engineering or medicinal research are pushing the boundaries to go beyond what we know about math to what it might yet be able to do.

Thomas Edison said, “To invent, you need a good imagination and a pile of junk.”[4] While all of modern mathematics may arguably not be a pile of junk, it is a pile of rules that we can learn to animate in different ways, with different voices. What it might be able to tell us that we have not yet considered?

Imagine the possibilities.

[1] NPR. “Brian Greene on The Hidden Reality.” Talk of the Nation, March 4, 2011. https://www.npr.org/2011/03/04/134265287/brian-greene-on-em-the-hidden-reality-em#:~:text=GREENE%3A%20I%20think%20math%20is,go%20to%20the%20next%20step%3F.

[2] “Relativity | Definition, Equations, & Facts.” 2023. Encyclopedia Britannica. November 7, 2023. https://www.britannica.com/science/relativity/Special-relativity.

[3] Sutter, Paul. 2018. “Relativity: The Thought Experiments behind Einstein’s Theory.” Space.Com, June 18, 2018. https://www.space.com/40920-relativity-power-of-equivalence.html.

[4] Preezly, Automatons of Yesteryear |. 2011. “‘To Invent, You Need a Good Imagination and a Pile of Junk.’ – Smithsonian Libraries and Archives / Unbound.” February 11, 2011. https://blog.library.si.edu/blog/2011/02/11/to-invent-you-need-a-good-imagination-and-a-pile-of-junk/#:~:text=When%20one%20thinks%20of%20inventors,Panama%20Pacific%20International%20Exposition%2C%201915.

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