There are a number of ways in which ambiguity comes into play in life, and its existence allows for creativity, excitement, innovation. It also allows for frustration, anger, and disillusionment.
In teaching, ambiguity can, when applied appropriately, be a powerful, positive force. But it can also lead to significant management or outcome issues if not recognized or ignored.
In this post, I want to address the problem with “ambiguity of goals” in courses.
In different disciplines, the goals can be shockingly vague: “write well” or “understand algebra.” Standardized tests can help clarify expectations, but even here the goals are rarely explicit: “do well on the calculus AP”; “do well on the US history SAT2.” Rarely (in private schools, at least) are specific targets on external tests given: generally, administrators simply look for “comparable performance.” Although not always clearly articulated, “comparable performance” usually means “compared to years past” and “compared to other sections of the same course.”
In courses where no standardized test is used as a metric, and where teachers design their own tests and metrics and assign their own grades, you can see that the situation is ripe for a number of problems. The most basic one is “What are the goals of this course, who sets them, and who measures the degree of their attainment?”
Scarcely anyone in the English dept would argue with the proposition that “writing well” is a highly desirable outcome of each of our core courses. Since there are different kinds of writing, however, there can be disagreement as to what kind of writing is most important; there can be disagreement as to how much time other desirable goals (such as speaking well) should take away from the time spent on improving writing; there can be disagreement on whether students are expected to hit some common standard or simply to “improve” a reasonable amount.
In math, “understanding algebra” (or whatever course) is scarcely a goal with which one could argue. But there is–or can be–plenty of disagreement about what it means. Here are some basic questions to which any math course (we’ve moved beyond the algebra example cited above) should, I think, have answers before one starts to teach it.
- What kinds of problems should students be expected to solve?
- What is the role of proof in the course?
- What kinds of creativity are expected/allowed in the course?
- How significant should applications of the subject to problems in other disciplines or “the real world” be?
- What kinds of methodology are to be emphasized in the course?
Let me give a few examples:
What kinds of problems should students be expected to solve?
Are the problems essentially single-step or multi-step? Do they require knowing only one basic concept or more? Are they identical or very similar to problems already done? Are they algorithmic, or should students have to think about how to solve them even if they “know the material”?
What is the role of proof in the course?
Are proofs and derivations basically ignored? Does the teacher present them during exposition? Are students expected to be able to reproduce given proofs or derivations on assessments? Are students expected to do proofs or derivations other than ones they’ve already seen?
What kinds of creativity are expected/allowed in the course?
As long as students learn the indicated algorithms/procedures and learn to recognize when to use them, will they do well? Are different, but mathematically sound, approaches to the same problem presented by the teacher or encouraged in the student’s work? Are there open-ended questions or discussions where students can more easily deviate from a “typical response path”? Are there independent projects in the course where students have some control over what problems they attack as well as on how they attack them?
How significant should applications of the subject to problems in other disciplines or “the real world” be?
In a “pure math” course, the question of applications by definition does not arise. But even number theorists can no longer claim such applications do not exist. The most they can now do is refuse to entertain them. Most students will be more interested in courses where they see some connection to the world around them, especially the parts of it in which they are interested. For your math course, is it a “pure math” course? (It’s extremely unlikely to be if it’s in high school.) If not, are the applications well thought out to develop mathematical thinking? Are they introduced at the beginning of a concept in order to motivate its study? Or are they thrown in as after-thoughts? Are they examples in which students might actually be interested? Are the necessary to understand, or can an uninterested student just tune them out? Do they tie in with the subject matter of other courses in which the students are enrolled? Do they presage some other math topic that will be coming up soon?
What kinds of methodology are to be emphasized in the course?
There are many levels at which any subject, math included, can be understood. The first is “not at all.” Presumably, that’s not a desirable level in a math course. But even here, “not at all” is scarcely accurate. Most people understand the basic math principles of counting, ordering, and grouping. It wasn’t until Russell and Whitehead tried to put math on a “first principles” basis that most people (at least, of those who tried to read their work) understood how incredibly profound those concepts are….
Another level is “use as a tool.” This is the level at which most people today understand their cars. They can turn them on, accelerate, brake, use them to solve problems that are nearly all variations on the “how do I get from location X to location Y” variety, and little more. Many math courses are mostly satisfied with this level of understanding, which is essentially algorithmic. “To get from here to there, get in the car, start it, see if backing out is required; if it is, put the car in reverse; if not, put it in drive…etc.” Some teachers, traditionally labeled “old school”, want students to understand how the algorithms they are using were derived or why they work. Others, also sometimes labeled “old school” want students to perform all calculations involved by hand without assistance from calculators or computers but do not insist on any deeper understanding of the algorithms employed.
Other teachers focus only on answers, on the assumption that if you don’t know what you’re plugging into the calculator, you won’t get the right answer. (This assumption is usually, but not invariably, true, even when all students are being honest. I periodically have students make offsetting mistakes and thus end up with the right answer through a flawed process. More common is the student’s ending up with the right answer coincidentally because the specific problem given happens to give the right answer with a flawed method of solution even though the same kind of problem with “different numbers” would not have done so).
In the free-response sections, AP exams in math tend to require students to show their reasoning even with a correct answer if they are to receive full credit.
And finally, as a harbinger of the fully connected world, you now have programs such as Wolfram Alpha which will often provide answers to amazingly complicated mathematical questions without the questioner seeming to have to know much if anything about the underlying “math.” Many teachers aren’t yet even aware, except in a vague way, of what these programs can do, and of those who are, many find them anathema.