"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The outdated mindset about the purpose of mathematics that many students bring with them when they transition from school to college is not the only problem many have to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.