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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Problem Solving … and Learning Mathematics

Our Math – Applications for Living course is sometimes used as a last option; students try passing the intermediate algebra class, and (after 2 or 3 tries) an adviser says that they have another option.  This is not true for all students in the course, though it is a common path to my door.  The result is a class with some very anxious students, and many who doubt their ability to solve ‘word problems’.

Math – Applications for living is all about problem solving; all topics are verbally stated.  We had an interesting experience last week when we did an example with a simple statement:

The distance from the Moon to the Earth is 3.8 x 10^5 km.  A light-year is 9.5 x 10^12 km; in one second, light travels 3 x 10^8 meters. How long does it take light to travel from the Moon to the Earth?

The problem presents to issues to resolve: the operation to perform, and making the units consistent (meters and km).  A few students knew to divide distance by speed to get time; if they did not already know this, it did not help much to solve the D=rt formula for t.  We explored the problem by working with rates (as we have been doing for most unit conversions); this helped a little more.

We got frustrated, however, with the km and meter conversion in the same problem.  After about 10 minutes of discussion, some progress was made.

In working through these struggles, more than one student said something like:

Can’t you just show us how to solve these in a way that we already understand?

Of course, it is exactly this gap between current understanding and present need that causes learning to happen.  As a problem solving issue, this is essentially a statement of what problem solving is … as opposed to exercises.  In the most encouraging manner, I told the class that this tension they are frustrated with — is the zone where we will learn something.  I stated, with emphasis, that if I did not create situations where there was a gap like this that they would leave the course with the same abilities as when they started.

I’ve been talking with faculty in some other programs at my college about the mathematical needs of their students.  The first thing they say is always ‘problem solving’, and they don’t mean solving a page of 20 ‘problems’ using the same steps.  The second thing they say depends on their program, and a surprisingly large number of them say ‘algebra’ is the next priority — in spite of the fact that algebra is often de-emphasized outside of the STEM-path.  In the Math – Applications for Living course, we use algebraic methods when useful, as it is when solving problems with percents.

In the larger context, all learning is problem solving.  A learner faces a situation where existing knowledge is not sufficient, and the gap is completed by some additional learning.  I believe that this statement is true regardless of the pedagogy a teacher uses, whether active or passive for the learner.   I do not agree with a constructivist viewpoint, especially the more radical forms; however, there is a basic element in the constructivist view that is true, I believe — knowledge is built as a result of gaps.  I believe that teachers can (and should) model the process of filling the gaps, and explaining the reasoning behind ideas that can help.  Learning math does not need to involve students stumbling through to discover centuries of mathematics; we can both guide and be a sage in the process.

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Some of My Best Friends are Calculators

Some years ago, we had an extended discussion about college credit for developmental courses (math in particular).  The proposal being discussed was eventually superseded by other policies; however, strong opinions were voiced.  During one commentary, a colleague was decrying students getting credit for such courses (though he had nothing against faculty who teach them.  Our Divisional Dean leaned over to me and said “some of my best friends are developmental math teachers”, which I thought was quite funny (though the situation prevented me from laughing at the time).

When I hear some colleagues talk about calculators, I am reminded of that comment rephrased … “some of my best friends are calculators”.  Calculators have their place, such colleagues say; calculators are not bad … it’s how students use them, so we need to prevent students from using calculators in a math class (as they say).  In fact, I once took the position that graphing calculators not be allowed in a first algebra course (back in 1993).  Since 1995, I have taught in an environment where graphing calculators are required starting with our first algebra course; although there are days when I find this frustrating, I have become a supporter of using calculators.

Unfortunately, the problem is much more complex than a ‘no calculator’ policy could solve; nor does a ‘required calculator’ policy solve these problems.  Here are some of the problems that we can avoid discussing by focusing on a calculator policy issue:

1. Students want a calculator for basic operations for a reason — they feel ‘dumb’ at math; that’s a major issue.
2. Students view correct answers as being a valuable commodity, instead of seeing correct answers as suggesting good understanding
3. Numeracy leads to feeling smarter; having a sense of how quantities ‘behave’ is possible for almost all humans (just like language literacy).
4. Reasoning about quantities is a natural human endeavor, though we communicate this with language systems that are artificial (a necessary condition)
5. A single math class tends to be very ineffective at changing long-held beliefs and habits; data suggesting an impact normally are measuring temporary conditions.
6. The big picture ideas are more important than how a student calculates a particular value; the big picture includes their self-image about mathematics.

I like requiring a calculator in math classes, to provide a better venue to discuss these issues with students.  Sometimes, a student ‘gets it’ (what we are talking about) and they change their math trajectory; for most students, it’s not that much of an issue either way — it took them 12 or more years to get to this point, mathematically, and a short-term experience is not likely to hurt them any more.  Using the calculator, it seems, at least opens the doors to possible positive changes over a longer period.

This conversation with myself started when somebody reminded me of an article I wrote for the 1993 AMATYC journal; reading that article was an awkward experience, as I could see errors in my own thinking.  Perhaps this post will encourage readers to examine their own position on calculators in math classes from a different perspective, one reflecting my course correction on the use of technology in mathematics.

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Important Things First … And Repeated

Like most faculty, I encounter times in the semester when I have to wonder “how did we get to this point?” — such as when a student in a course like intermediate algebra does not recognize a product versus a sum, or can not recognize a right-triangle distance problem in context.  I could follow the path of blaming previous bad teachers (all of them except me [:)] ), or on students who do not study; there might even be some truth in these explanations.  However, the professional response is to explore how my course enabled these problems to survive until the end of the semester.

I am concluding that we (and I) stop working on ‘basics’ too soon; I (and we) presume that a passing score on an assessment like a chapter test shows that a student has the basics.  However, I suspect that I depend too much on closed-task items on assessments, which enables some students to simulate appropriate knowledge without its presence.  In addition, I am concluding that I need to design classroom interactions to constantly build literacy and analysis of mathematical objects.

People often say ‘mathematics is a language’, and promptly teach mathematics as if it was a set of mainline cultural artifacts.  We can learn much from our colleagues in foreign language instruction, who tend to constantly use basic literacy into all work in a language and to deliberately address the cultural components of the language.  I see most of my student’s basic failures within mathematics to be cultural issues (context, norms) along with language literacy within mathematics.

The implication I see for my own teaching is that classroom time needs to deal with ‘sum or product’ as an issue every day; nothing is more basic than this issue.  In algebraic classes, there is an added layer of work on symbols and syntax which needs a similar focus (sum or product).  I’m also seeing a need to deliberately address reading skills applied to a math textbook, and hope to coordinate these types of efforts.

I am constantly reminded of this notion:  Novices do not automatically see the critical features and structures that experts see without effort.  Our students are capable of more, and can reason mathematically.  We need to deliberately show the features and structures we see, and provide scaffolding for students to become more expert.  We do students no good if they leave a math class in the same novice mode as they started, with some limited problems they can solve.

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