Technology in Mathematics:What We Gain and What

We LoseDerek Thompson

Taylor University SNAS Seminar October 27, 2014

Technology in Mathematics Teaching◦ How we teach◦ What we teach

Technology in Mathematics Research My One Wish Mathematics as Worship

Outline

Sad/Funny story… (“Can I use my calculator?”)

The obvious battle: calculators / wolfram|alpha◦ Arguments against

We sacrifice material – students aren’t learning things that they would have before

Students aren’t as aware of the theoretical side Students don’t gain the insight they would from hard

problems they were previously asked to do How can we innovate if we don’t fully understand

what we already have?

How we teach

Some more “sad” stories Volume of a cylinder: 1 - √3 + √5 - √7 + ….

◦ He didn’t have the intuition that comes by working these out by hand. He should know this can’t be what a right answer looks like.

My calculator just tells me sin(∞)/ ∞ = 0…◦ The TI-89 is sometimes TOO good; the student now

thinks sin(∞) is an object he can work with. It’s nonsense!

◦ Quick other thought on Wolfram|Alpha: its parsing of language is also SO good that it may hurt students in the long run, letting them think bad syntax is OK

How we teach

Calculators / wolfram|alpha, continued◦ Arguments for:

This is how they will work problems in their career We can do problems more intense than what could

be reasonably done by hand Technology can motivate students to learn, instead

of motivating them to take the easy way out See next slide!

How we teach

While we want students to learn syntax, calculators could be certainly be more intuitive than they are – without giving things away, or giving false insight.

Derivatives, sliders, etc. in Desmos Students taught me things here!

How we teach

Students are “Emerging Adults” Colleagues’ College sounded negative overall These students stare at screens and phones

all day… Well, let’s use that! These students are great at playing in digital

sandboxes Technology has great power to motivate, but

we could use it better. TI-89 is over 15 years old…

How we teach

Mastery and Gaming Are completion grades appropriate? At what point do we need to start deducting

points? Can we have “healthy confusion” move to

understanding without punishment in between?

Should students really “Start with an A” and go downhill from there?

How do students achieve success at video games?

How we teach

Online homework Isn’t perfect, but many advantages Immediate feedback Desire to get it right Immediate gratification / sense of accomplishment Badges? Points? Gamification of grading? Why not?

◦ XP could “hide” traditional grading scale but be the same◦ Online homework would allow for “grinding”

They think they’re cheating the system, but they aren’t (personal anecdote)

Rewards: homework skips, extensions, test retakes, etc.

How we teach

Some more thoughts on video/board games, and play in general:

The far extent of my vocabulary came from Magic: the Gathering.

My hand-eye coordination came from Mario Bros. My quickness of thought came from Street Fighter II My puzzle-solving skills came from Legend of Zelda Explaining board games has taught me quite a bit

about teaching! How do children learn? Through play! Technology lets us keep that sense of exploration

longer

How we teach:

A source of confusion: It seems that we confuse our frustration

with “gimme” culture with technology, and blame the technology for hearts that don’t want to learn.

“I just want to know how to get the right answer”

Yet, your boss will demand that you think creatively…

How we teach

“First grade” problem

The problem seems to stem from the fact that technology easily solves yesterday’s problems that required creativity (example: integration).

Who do we blame for low test scores – students, parents, government? …Ourselves?

Maybe the problem is that we’re still asking those questions? Can’t we use technology to ask new problems that still require creativity, even if technology is involved?

Personal struggle with Hughes-Hallett

How we teach

What do want our math majors to know? Is there a market out there for a pure mathematician? Which of the following is not like the rest:

A. An engineerB. An actuaryC. A pure mathematicianD. A large pizza

The answer is C. The rest can feed a family of four. Our math majors need to know applied mathematics,

and know it well. What does that look like? Is Calculus still the center of everything? I don’t think so!

What we teach

Carl C. Cowen, 1997, “On the Centrality of Linear Algebra in the Curriculum”

“...the first realization that we must come to is that linear algebra is incredibly useful in the modern world, probably more useful than any other college level mathematics with the possible exception of calculus. ”

“The second realization we must come to is that no serious application of linear algebra happens without a computer.”

What we teach

Linear Algebra (1950’s!)◦ Every spreadsheet is a matrix◦ Engineers, physicists use eigenvalue problems◦ Everything a computer does is discrete

Linear Algebra bridges that gap between discrete mathematics and continuous mathematics

◦ Segues to other CS topics: graph theory, Markov chains, etc.

Other new foci: discrete mathematics, numerical analysis

Taylor is doing great at these things (AFAIK)

What we teach

As we add emphasis on ‘new’ mathematics (linear algebra, numerical analysis…), do we add more credit hours? Or do we take away ‘old’ mathematics?

How do we know what’s important? We don’t!

What are we sacrificing?

Considerable work in number theory (1877-1947)

“Mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose very remoteness from ordinary human activities should keep it gentle and clean.”

“Nothing I have ever done is of the slightest practical use.”

Nowadays, number theory is used in computers every day!

Quotes from G.H. Hardy

Technology has changed our content focus◦ But we have to be wary of what we sacrifice◦ And the importance of higher-order thinking

It has changed our learning methodologies as well◦ We might be resistant to that, and in some ways,

maybe we should be◦ But we can also embrace it in ways we haven’t

yet

Teaching Summary

Graduation story at IUPUI◦ My own “G.H. Hardy” moment◦ Is it OK for research to only serve other

mathematicians (immediately)?◦ “Used in lasers” – uh huh…

Technology in Mathematics Research

Capable of much more than ever before Very high computation can give guidance to

theoretical results, though it cannot replace theoretical work (100x100 matrices, etc.)

Computer-assisted proof◦ Mere evidence is never a proof, however…◦ Controversial, but also fruitful

We could get farther if not for a lack of cooperation….

Technology in Mathematics Research

If I had one wish, it’s that we use technology to “clean up” mathematical language and break barriers for students

Awful notation in mathematics:◦ 1 divides the norm of the absolute value of 1:|||||||||◦ “Does not divide” looks like “times” looks like “cross product”◦ (1,2) = x-interval or (x,y) point?◦ Inverse notation cannot be ‘touched’◦ The list goes on…

What if Apple was in charge of making mathematical notation?

Aren’t we at an age in history where we shouldn’t stay shackled to “the way things have always been”?◦ Most of this notation is centuries old…

Where can technology take us from here?

Mathematics is unique among the sciences◦ I didn’t “create” or “invent” a theorem, I

“discovered” it◦ Was the Pythagorean Theorem true before it was

written down? We find a basis for universal truth We also find a sense of God’s mystery

◦ Non-integrable elementary functions, 0! = 1, theorems that are undecidable

◦ Truth transcends knowledge

Mathematics as Worship

When teaching p -> q, I describe it as a “promise”

Are theorems, then, promises of God? Seeking them, then, is seeking the wisdom

and knowledge of God What’s that got to do with technology? Technology lets us dig even deeper and

seek God’s kingdom further, letting us uncover even more truths about the universe

Mathematics as Worship

drthompson@taylor.edu

Thank you!