"Correct answers are essential... but they're part of the process, they're not the product. The product is the math the kids walk away with in their heads." —Phil Daro

If you have not already watched Phil Daro's 17-minute video Against Answer Getting, you should do so
right away. (I'll keep this post short to give you enough time to watch it in
its entirety.)

Daro, a
longtime mathematics educator and leading figure in the national mathematics
education community, is currently director of the San Francisco field site of SERP, the Strategic
Education Research Partnership. He was one of the mathematics educators who
played a leading role in the formulation of the mathematics Common Core State
Standards. (You know, one of those knowledgeable experts the StopCommonCore
brigade keep claiming were not involved in CCSS development.)

The video is full of powerful insights that the mathematics
education community has accumulated over many years of research. My opening
quote sums up the focus of the video. Here is another one I like:

"Mathematics does not break down into lesson-sized pieces." —Phil Daro

This particular quote resonates with me. I adopted the same principle in
the design of my MOOC Introduction to Mathematical Thinking, currently about halfway through its fifth run.

Daro's focus, both in the video and in his work in general, is K-12
mathematics education. But it is very relevant to those of us in college-level
mathematics education. When students come to college with a perception that
mathematics is about "answer getting," we face the very uphill task of ridding
them of that misleading mindset.

True, for hundreds of years, getting answers was a key component of
learning and doing mathematics. But these days, if we want answers in
mathematics, we generally use one of a number of digital technologies. The job
of today's mathematician (or typical user of mathematics) is problem solving.
The part that requires a human mind is when the problem has a novel aspect. It
was precisely to put the focus on the thinking part that I named my MOOC the
way I did.

The principle requirement for being able to solve a novel problem is
conceptual understanding. That is why the issues Daro raises in that video are
so central to the mathematics education of the citizens of tomorrow.

The outdated mindset about the purpose of mathematics that many students
bring with them when they transition from school to college is not the only problem
many have to overcome. A parallel issue manifests itself when they start to
learn about mathematical proofs (if they follow the mathematics path).

My MOOC students are currently right in the middle of that part of the
course (proofs), and many are having a very hard time coming to understand what
role proofs play and what (therefore) constitutes a good proof.

The dominant perception is that proofs are what mathematicians produce
in order to determine mathematical truth. That, of course, is true (at least in
an idealistic sense that guides mathematical progress), but as with arithmetic
answer getting, it is only part of the story. And in terms of actual
mathematical practice, a very small part of the story.

As with answer getting in K-12 math, achieving a logically correct proof
is a binary target (right or wrong), which make both very easy to evaluate for
correctness and assign a numerical grade. (Ka-ching!)

But let's pause and ask ourselves how proofs work in practice. If you
want to know if Fermat's Last Theorem is true, you consult a reliable source.
Today, any moderately knowledgeable mathematician will tell you the answer: "Yes." Now you know.

But what if you want to know

*why*it is true. That's when you need to look at a proof.
In terms of mathematical practice, proofs are about understanding. They
are communicative devices we construct to convince ourselves and to convince
others.

In my MOOC, because I cannot assume the students have access to individualized,
expert feedback on their work, I do not ask them to construct proofs. But I do
present them with a range of purported proofs, some correct, others not, and
ask them to evaluate them. The evaluation is in terms both of logical
correctness and communicative effectiveness.

To do this, I ask them to look at each purported proof in terms of five
different factors: one logical correctness, the others focusing on
communicative issues. Though the five factors are not independent variables, I
ask them to treat them as such when evaluating a proof.

This is the part of the course where those students who have had some
exposure to proofs in the K-12 system tend to do worse than those who are new
to proofs. They are simply not able to approach a proof other than in the "answer getting" mode of "Is it logically correct?"

This shows up dramatically with extremal cases. When I present them with
a carefully constructed argument that is logically correct but provides no
explanation, they will give it high marks across the board. But faced with an
argument that is superbly articulated but has a logical flaw, they are
psychologically unable to evaluate the structure of the argument. "It's wrong," they keep saying. End of story (for them).

Of course, extremal examples are atypical, and often difficult to wrap
our minds around. That's what makes them so valuable as learning devices. It's
when the classroom rubber hits the road and we find ourselves using
mathematical thinking in our lives or careers that it becomes important to have
good communication skills.

Pick up a more advanced level mathematics book or research article and
the chances are high that the arguments presented will contain errors.
(Actually, the book does not have to be advanced. Euclid's

*Elements*is littered with "proofs" that are not logically sound.) But if the arguments are well laid out, with adequate explanations, a suitably skilled reader can fix them as they go along—possibly with help from someone else. (That's definitely the case with*Elements*, though it took two thousand years before David Hilbert noticed that Euclid's own arguments left a lot of work to be done to make them genuine "proofs.")
It's the same in software engineering. Any useful program will have
bugs. But if the code is well structured, and adequately annotated, someone
else can dive in and fix it whenever a flaw manifests. A

*good*computer programmer is not someone who writes error-free, working code; it is someone who writes working code that can easily be fixed or modified.
I'll leave it as an exercise for the reader to identify the analogous
issue in the natural sciences.

If those of us in the education business want to do the best we can to
prepare our students for life in the 21st century, we need to recognize
that in an era when technologies provide instant answers (facts), the one
ability they will need above anything else is (creative, reflective) thinking.