Towards Effective Remediation: Culture of Learning

Two events are fairly common in my classes: (1) A student says “just tell us” when we are doing group work, and (2) A student says “I don’t get that at all” when I am doing a (mini) lecture.  I suspect that these are also common in other classes, and wonder what meaning we each see in these statements.

Both statements deal with confusion and frustration.  One occurs when a student is struggling to find the idea in their own work, and the other occurs when a student struggles to understand the idea in my work.  Both are normal, and both are part of a learning process.

A culture of learning would be shown by an acceptance of these frustrations, combined with a determination to learn in spite of (or because of) that frustration.  Learning is rewarding just for its own sake as we see how ideas connect and build on each other.  A focus on comfort defeats a learning attitude.  Perhaps a focus on the learner raises the same risk.

We tend to see the phrase “student centered” as a positive goal usually implying a process whereby students find ideas about mathematics.  For some of us, this means that we seek to minimize frustration and/or confusion.  I think a better goal is to manage the frustration and confusion to maximize learning and build a culture of learning.  I want my students to see learning mathematics as a set of goals which are attainable given effort and attitude.

We can also see ‘student centered’ as an idea leading to a focus on context and applications, perhaps to the extent that we only cover mathematics that can be applied to problems of interest to students.  As much as I am enthusiastic about applications (I teach a course 100% ‘applications’) I think it is a mistake to construct a curriculum around problems that students can understand and care about — these must be included, but a culture of learning means that we look at extending beyond the immediately practical to the larger ideas and even the artistic beauty of the subject.

In every course, I seek to present some beautiful and useless mathematics.

I know that few of my students achieve this culture of learning, even though my goal is to get them so motivated to learn that nothing will stop them from learning more mathematics.  I know that most of my students will stop taking mathematics as soon as that becomes an option, even though my goal is to inspire them to take at least one more math course than they are required to take.

Students seldom achieve more than our goals and expectations, so I have this culture of learning as a goal in my classes.  Rather than a limited range of ‘student centered’ ideas, I am looking at the largest possible picture of what that means — including how we deal with frustrations and confusions.  Learning, as in life, mostly is determined by how we deal with such problems; learning, as in life, is damaged by attempts to avoid confusion and frustration.

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2 Comments

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  By schremmer, February 15, 2013 @ 3:21 pm

  I agree with much of what Professor Rotman is saying about confusion and frustration but I think he ignores another aspect of the issue.

  In order for students to acquire a bit of the culture of learning (mathematics), they have to deal with mathematics the way mathematicians do. In particular, they should deal with the *construction* of mathematics. The question, then, is to decide on what view to take on what mathematics is about. I prefer to look at mathematics in a model-theoretic way that is as a paper-world representation of the real-world but one could certainly look at mathematics as just a paper-world game. Either way, mathematics is constructed and even so-called developmental students can take pleasure in such a construction. For instance, in constructing an arithmetic representation of the quantitative side of the real world. Starting with plain number-phrases to represent collections of like items, continuing with signed number-phrases to represent collections of like items moving this or that way. Decimal number-phrases on the other hand are needed both to deal with large collections and with “amounts”. One could want to go into multi-arithmetic by constructing a system to represent “shopping carts” and “price lists”. Etc

  –schremmer

  P. S. As for myself, I try to diminish the number of students who “will stop taking mathematics as soon as that becomes an option” and so, it is quite possible that I am overdoing this aspect that Professor Rotman seems to be ignoring. Still . . .

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  By Diane W., February 27, 2013 @ 6:10 am

  I liken learning anything to muscle development. You have to “feel the burn.” You must do some grappling with a truly new concept — especially one that challenges your presuppositions — in order to learn it and incorporate it into your own schema. A brief period of confusion is normal and to be celebrated. It means we’re going to come out on the other side with an improved understanding.

  When students become discouraged, I tell them that during their sleep that night their brain will work on incorporating this new understanding. We will re-visit it tomorrow, and it won’t seem so daunting. I then plan out several days of gentle and brief reinforcements of the new material.

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