How do students view mathematics?
Not surprisingly, many (if not most) students see mathematics as a set of known problems to be solved — changing a few numbers here and there, perhaps — but essentially, all of mathematics is known.
Mathematicians have rather the opposite view — we’re just scratching the surface. There is so much more underneath.
As I mentioned in my first post, mathematics is creative. What makes this difficult for students to appreciate is that the artistic medium is that of abstraction, and without a real understanding of abstraction in mathematics, the creative aspect is hard to see.
But there is a way to help students experience the creative side of doing mathematics — and that is by having them write their own Original Problems. I began thinking about this while I was teaching a course in problem solving at a magnet STEM high school — and being an avid problem writer myself, I imagined that having students write problems would help them solve problems. Whether this is true or not is difficult to determine. Regardless, an assignment was born….
What really got me interested in this assignment was the student comments at the end of the semester. One student wrote,
Anyone can write tedious, difficult problems that review core math subjects, but to write problems in a novel, challenging, and refreshing manner, one must be imaginative. I feel that this creative side of math is an often overlooked aspect of the field as many believe math to be an extremely black-and- white, rigid, and boring subject.
I was intrigued by the fact that even though students were not prompted to address creativity in writing their course evaluations, some spontaneously did so. As a teacher, I was delighted — an unanticipated side effect of an assignment designed for another purpose was somehow more significant to me than the intended outcome.
Fueled by this success, I introduced the assignment in an Honors Calculus section I taught, and students responded positively again. Then I incorporated writing Original Problems into a traditional calculus classroom, then precalculus, then algebra — and students kept getting it. Posing problems was no longer an assignment just for advanced students.
What does the assignment look like? I break it down into four sections. First, Motivation. Where did the problem come from? For some students, they might start looking in their textbook at interesting problems. For others, they take inspiration from their daily life. One calculus student said he came up with his problem because he dropped his backpack down the stairs and had to retrieve it — and he immediately thought of this as a displacement/velocity problem. Another student was doodling figure eights, and created a problem about ice skating on a figure-eight shaped rink. It is remarkable what students can create, given the opportunity!
Second, the Problem Statement. This is actually quite difficult for some students. And we teachers know the challenge of writing a test whose problems can be interpreted in only one way. Now that I’ve moved on from the STEM high school to teaching university again — and work with a different set of students — I now assign the Motivation and Problem Statement as a separate assignment. That way, I can give written and verbal feedback to students and help them craft a well-stated, manageable problem. This has been very helpful for the students, and the quality of their final submissions has improved.
Third is the Problem Solution. This is fairly self-explanatory, but a few comments are in order. I like to give students wide latitude in selecting a problem of interest to them — sometimes they want to challenge themselves with a difficult problem. In this case, a partial solution is fine. The point is to get them writing mathematics — and a partial solution to a difficult problem often involves more mathematics than a complete solution to a more routine problem.
Finally, there is the Reflection. I only ask for a few sentences or a paragraph — enough to give me a sense of how students are responding to the assignment. These can be very revealing, and you sometimes get students who really appreciate the assignment and understand its purpose. All four sections are to be included in the final submission.
You might be interested in a recent Original Problem prompt. This assignment is highly adaptable. I’ve had colleagues who wanted to narrow the focus because the assignment seemed to broad. Suggesting a specific application — such as the Pythagorean theorem — will give students a starting place. In my mind, the assignment is about creativity, writing, and self-determination. Let students choose a topic to create a scenario and write about it, and they start to get a handle on what creativity in mathematics is all about. There is no one way to accomplish this.
I should say a few words about grading these assignments. At their broadest, these assignments read like short essays. But they’re all different, so you can’t really develop a rhythm in the grading process. So Original Problems take more time to grade — this semester I’m just giving two assignments, so it’s more manageable. I do think it’s important to give at least two assignment, so students have a chance to improve. Generally, I’m more lenient when I grade the first assignment, since often this is the first time students will have encountered such an assignment.
To encourage creativity, I tell students that if they just do the assignment — and get their mathematics correct — they won’t earn lower than a B. I don’t want them worrying about grades (and we’re stuck with them for a while!), although some inevitably do. I rarely give a C, unless it’s evident a student waited until the last minute, or a student worked below their potential. I do believe that for an assignment like this, you should evaluate students relative to themselves, not their peers. More able students should be pushed — and frankly, most of them appreciate it when you do push them.
Many students really do begin to understand the creative aspect of mathematics after doing these assignments. They really do enjoy getting to choose their own problem — and though it is sometimes challenging to come up with a way for them to develop a particular idea, I rarely tell them to just choose another topic. I try to find some avenue they can pursue.
So I encourage you to give Original Problems a try! Let me know how it goes. For additional reading, you can find an article about writing Original Problems in Publication 10 on my website. There is also a discussion of several student problems in Chapter 6 of Mathematical Problem Posing. It really is time to have all students experience creativity in mathematics. This is one of the main purposes of writing this blog, after all.