Should a Math Class be an Approximation?

I was tempted to title this post “should high school math teachers be allowed to teach college?”, because that is what I was thinking about recently; of course, that is not really the issue.  The issue is: should a math class, such as developmental math, be an approximation of the mathematics or should it be precisely the mathematics?

Here is the situation that got me thinking about it … While I was at the AMATYC conference (and Dev Math Summit) last week in Anaheim, I had substitutes in all four of my classes.  Two of them were former high school teachers, one a retired state worker.  Both of the former high school teachers confused my students by their mathematical presentation.  In one case, the teacher said that students should shade all regions for each inequality in a system including the ‘double overlap’; this is an approximation to the mathematics — only the overlap area should be shaded.  In the other case, the topic was the imaginary unit and complex numbers; this teacher did not appear to say anything ‘wrong’ but focused entirely on the mechanics.

It’s probably obvious that no math class can achieve precision in all topics during the learning process.  Approximations come from various factors, some more malleable than others.  One factor is linguistic in nature … precision is based on language, and deep understanding of language comes with experience.  We can not expect an expert understanding of the language from novice users; however, I would like to think that we design courses and curriculum so that students will move steadily towards the expert level.  This is complex, perhaps impossible … but I think it is critical to invest energy in this process for students.

Another factor for precision is created in the modeling process we provide to students.  In the ‘imaginary’ number case last week, the instructor emphasized correct symbol manipulation as a proxy for understanding the topic.  However, the human brain does not store information in a purely symbolic form … the process involves a verbal statement (sometimes called ‘unpacking’) from the symbols.  A novice student has no knowledge connected to the symbols; my substitute confused students by not supporting a verbal (conceptual) framework.

To re-state the title …

Should a math class be a deliberate or accidental approximation?

At this point, we should be thinking something like “Well, what is the problem if a math class is an approximation (deliberate or otherwise)?”

Here is a key problem:

Correcting prior knowledge is more difficult than creating accurate knowledge.

You may have noticed that students’ understanding of fractions is resistant to our efforts of ‘correction’ (same with algebraic faux paus such as distributing a power over a sum).  We spend millions of dollars on instruction partially as a result of math classes being a approximations at some prior stage(s) of the student’s math history.  Every time a student is required to take a standardized test in math, we are seeing the direct results of approximations in math classes and the harm they cause students.

I have no delusions that excessive ‘approximations’ are limited to K-12 teachers; I’m sure that many of our college instructors and professors do the same kinds of things.  I am guessing, however, that school teachers are more prone to running approximate math classes (based on interviewing experienced teachers across levels).  Also, policy makers who focus on ‘skills’ often provide indirect motivation to make math classes more approximate, as does a focus on ‘teaching for the test’.

Here is the tension we face:

Approximations result in inaccurate or incorrect learning.

Perfect precision results in no learning at all.

This is the math teacher’s paradox.  Like most paradoxes, this one serves to sharpen our problem solving.  The solutions lie along a path where approximations are deliberately limited and then refined towards perfection over time.

Many of us seriously underestimate the amount of work needed to learn mathematics — both professionals and policy makers.  Resolving the math teacher’s paradox depends upon appropriate conditions; the most basic of these conditions is time.  “Covering” the Common Core or “covering” the algebra curriculum will tend to doom most students to suffering the consequences of repeated approximations to mathematics.

We’d be better off working on precision for a lot less curricular content.

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AMATYC Presentations about New Life Math

To see the presentations and handouts for my sessions on New Life at the 2013 AMATYC Conference and Developmental Mathematics Summit, see the page AMATYC 2013 and Developmental Math Summit 2013    [also known as https://www.devmathrevival.net/?page_id=1807]

Probem Solving in a Digital Age

Whether we ‘flip’ a classroom, use an online homework system, or refer students to Khan’s Academy … our students are using task-oriented videos.  In addition, students have a tendency to see ‘look it up online’ as a substitute for learning something.  As we become immersed in (and dependent upon) the digital age, can we still work on problem solving or critical thinking?

One of the sessions at this year’s AMATYC conference dealt with the topic ‘stop the assault on critical thinking’.  In the session, they played the roll of a short video on subject (related rates in calculus, maximizing a function in pre-calculus, or unit conversions in a liberal arts math class).  The audience experienced something like a typical 3 minute video on that topic, and then we talked about how this supported critical thinking (or not).

The next session was one by Jim Stigler on ‘using teaching as a lever for change’, though he talked more about the futility of identifying specific teaching activities as being ‘effective’.  Dr. Stigler did include 3 aspects of teaching that are connected to improved learning — productive struggle, connections, and deliberate practice.  Learning is a complex process, and the presence of these 3 factors in the learning environment are connected to improved learning.

So, there is a connection between this research-based observation and the concern about critical thinking, I think:

Discrete learning experiences like short videos focused on successful completion of a task, based on clarity and being easy to follow, are guaranteed to limit both overall learning and critical thinking.

Mathematicians hold critical thinking as a goal to be valued; we want students to be able to flexibly apply knowledge to novel situations and interpret results.  This seems to be a basic problem.  Our students expect math to not make sense, that they could not figure something out; task-oriented videos support this self-defeating belief.

We can not hide from the digital age, even if we wanted to.  However, we can improve our understanding of the factors that contribute to the learning opportunities for our students.  A balanced approach appropriate to each course can help students through the learning process — including the struggle, the connections, and the deliberate practice.  We might even see these digital resources as just-in-time remediation to be used occasionally, rather than seeing the digital material as the basic course.

We need a more subtle understanding of how our teaching can contribute to student learning.  A single belief or methodology will not succeed for our students, no matter how good of an idea we might have.

To see a presentation by Dr. Stigler similar to what he did at the conference, see http://www.salesmanshipclub.org/downloadables/scyfc-Stigler.pdf

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