**FREGE**: **F**undamentals
of **Re**asonin**g** for the **E**lectronic Age

Clear thinking is required
for success in all human endeavors.
“Clear” is not equivalent to “formal” and none of us writes out a formal
logic proof every time we need to answer an important question. But formal logic is an important pedagogical
tool in much the same way that other “exercises” are:

·
Piano students practice scales.

·
Athletes do pushups.

·
Thinkers (of all stripes) write small proofs in formal logic so
that they understand the difference between sound and unsound reasoning.

But formal logic is more
than an exercise for students. It’s also
a powerful practical tool in its own right.
For example:

·
Mathematicians write formal proofs, both as a way to derive
insight and as a way to certify truth.

·
Computer scientists write formal specifications for programs and
applications. Then they reason with
those specifications to construct working systems.

·
Computer engineers use logic as the basis for the design of
computer circuits.

Yet, despite its
importance, logic shows up barely (if at all) in most pre-college curricula
today. At one time, most high school
geometry classes had a large proof-based component. But that is no longer true.

So students come to college
with almost no exposure to this fundamental tool. Our goal is to create an online learning
environment that fixes that problem.

There exist hundreds (or
maybe thousands) of books on logic and discrete mathematics. There are also already some online logic
classes. But we haven’t found any of
them that meet all of our goals. In
particular:

·
We want a focus on the practical use of logic as a reasoning
tool.

·
We want to help students from all disciplines become better
thinkers.

·
We want to grab the attention of students in disciplines in
which formal logic plays a particularly key role. In particular, we want to exploit concrete
examples from computing and mathematics.

·
We want to target computer science and engineering students in a
special way. Formal reasoning will be
critical in everything that they will do.
But they don’t always appreciate that.
Many think that they just need to learn to code. So we need to rope them in with useful
techniques that bridge a perceived abstract/concrete chasm. But, at the same time, we need to structure
the course so that those excursions are not off-putting to noncomputing
students.

So we want a course that
is laced with diverse examples, including:

·
The standard fare of logic courses: everyday reasoning. These examples will speak to technical and
nontechnical students alike. For
example, they’re good preparation for the law school entrance exam.

·
Mathematical examples that are not so trivial as to lead
students to say, “So now I can prove the obvious,” nor yet so advanced as to be
inaccessible to novice mathematics students.

·
Computer science examples that draw in and motivate not just CS
students but many others who have grown up in the digital age.

We also want a (smaller)
follow-on course that builds on the logic one.
Nontechnical students will likely choose to skip this second
course. But for technical students, this
course on Sets, Relations and Functions will:

·
Introduce those important ideas,

·
Provide additional practice at proving things once there are
interesting structures to prove things about.

While, in principle,
students could take an unrelated course that covers this material, in practice
notation can be a huge hurdle. Our
second course will use all the same notational and proof conventions as does
the reasoning course. That will let
students procrastinate, until they have a bit more mathematical sophistication,
jumping into new notational worlds.

There’s no single “best”
entry point into this large web of material.
So our goal is to build a system that can be used in many ways,
including as:

·
A prematriculation resource: We envision making it available to students
in the summer before they arrive at UT.
Our very specific goal here is to make students better prepared to
succeed in their first semester at UT.

·
A flipped class resource:
We are designing it so that it could be the basis for an efficiently-run
organized UT class that covers this material.

·
A just in time learning resource: We see it as something students can use when they
recognize the need to fill in gaps as they pursue their UT coursework.

·
(Possibly) the basis for a dual enrollment course that could
give potential UT students access to the material while they are still in high
school.

To make all of those uses
possible, we need to offer substantial flexibility in how and when students can
use our materials. So we rejected the
idea of a synchronized course (such as a MOOC).

Instead, our goal is to
build what we’re thinking of as an online textbook. It has:

·
“Picturated” text. Formal logic is, at its core, a symbolic
process. So text is more appropriate
here than it is for many other kinds of things.
But we can’t ignore the importance of visual learning. Fortunately, the online environment makes it
possible to include pictures and graphics much more easily and cheaply than
could be done in the offline world.

·
Video, when video is the most effective medium. In particular, videos are great for how- to
lessons such as, “How to Discover a Proof”.

·
Interaction almost all the time. No dozing allowed. Unlike
traditional books, and unlike even many video based online classes, we carve up
knowledge into very tiny morsels. After each morsel, students must think,
interact, explore and practice.

·
Flexibility. The material
is structured so that it is easy for students to run quickly through any
material that they’ve seen before. Then
they can focus their time on material that is new or difficult for them.