Avoiding Problems and Disasters in the Learning of Mathematics

In some implementations of reformed mathematics courses, there is a strong emphasis on particular ‘learning’ procedures in our effort to improve student outcomes. For developmental mathematics, the learning procedures du jour are problem-based learning and discovery learning (both heavily influenced by constructivist viewpoints). These methods have some basis in research and theory, but are easier to implement badly than well (as is true for most procedures). The purpose of this post is to suggest some guidelines that can avoid issues with these methodologies.

First, I recommend people read a summary of such methods by Krischner, Sweller, and Clark called “Why Minimal Guidance During Instruction Does Not Work” available at http://www.cogtech.usc.edu/publications/kirschner_Sweller_Clark.pdf . Here is their conclusion:

After a half-century of advocacy associated with instruction using minimal guidance, it appears that there is no body of
research supporting the technique. In so far as there is any evidence from controlled studies, it almost uniformly supports
direct, strong instructional guidance rather than constructivist-based minimal guidance during the instruction of novice to
intermediate learners. Even for students with considerable prior knowledge, strong guidance while learning is most often
found to be equally effective as unguided approaches. Not only is unguided instruction normally less effective; there is
also evidence that it may have negative results when students acquire misconceptions or incomplete or disorganized knowledge. [pgs 83 and 84]

Second, here is my own summary of other research:

Lectures are a poor method of providing direct, strong instructional guidance.

Our problem, and confusion, stems from a reaction to the lecture methodologies. Students tend to be passive and not engaged, so our reaction looks for methods that make activity visible. However, visible activity may be worse than a passive student (that’s what the research summary above says). We can not settle for what is easy to see; we must go beyond the ideology of ‘learner centered’ and focus on LEARNING.

How people learn is not that much of a mystery. For people with a brain functioning in the normal ranges, here is the recipe:

New information that does not fit existing knowledge
Effort applied to reconcile this gap or conflict
Access to information related to this gap
Validation of the resulting new knowledge

Of course, this is overly simplified to be prescriptive for use in a classroom. However, we need to keep our minds on these ingredients, not on visible activity. When discovery learning fails, it is often due to a design where the focus is on the first two ingredients; the mythical lecture mode involves a focus on the third ingredient.

I received a message from somebody teaching in a Math Lit course, who was frustrated by the difficulties students were encountering. To me, those difficulties originated from an almost exclusive focus on new information and effort; the message spoke to the missing information (step 3), though step 4 is an issue in some courses as well (practice and assessment). We can’t expect students — at any level of mathematics — to spontaneously create good mathematics that is integrated in their brain.

Yesterday, I had a brief conversation with my Provost after a professional development session emphasizing the advantages of a flipped classroom (which is a different issue than those above). When I told the Provost that I was not impressed with the presentation, the Provost responded with something like “That’s okay; we mostly wanted to get people moving away from lecture.” Yes, we need to move away from lectures; most of my colleagues have already done that. We also need to move away from the antithesis of lectures to models where students are active but not productive.

So, provide summaries and mathematical statements to your students. I don’t care whether this occurs before or after they attempt problems, as long as you design the experience for learning. And, provide instruction to your students; you are an expert, and they will tend to remain novices when they do not see how experts do the same work. Keep a large emphasis on validation and assessment; however students learn, they tend to store information either partially correct or completely incorrect, and it is our job to provide a rich diet of feedback on this learning. Remember that learning is never visible; what happens in a brain is hidden from most of us (who lack fMRI machines in our classrooms). And, be sure that you don’t confuse activity with learning.

Learning is not easy; designing a math class for learning is not easy. The ideas of learning are simple, but the application to a classroom is nuanced, requiring attention to all components of learning. Any instructional design that emphasizes a subset of ingredients for learning is going to fail students. In our rush away from ‘lecturing’, we need to avoid jumping off the cliff in to the lake labeled ‘active students’; this lake might look attractive, but being in the water does not mean there is learning about the water.

Join Dev Math Revival on Facebook:

cognition | Jack Rotman | May 16, 2014 10:33 am

1 Comment

By schremmer, May 16, 2014 @ 11:45 am

(1) A large part of the problem is that what is being taught is to mathematics what instant coffee is to espresso. Physicists realized that a long time ago. We have not.

(2) Mathematics has always be learnt through reading “pencil in hand” and then “discussion”.

(3) The problem with (not only) developmental students is that they are not used to learn through reading. It seems to me that coupling a (real) mathematics course with an English Reading course in which the reading materials were the (real) mathematics text ought to work well.

Regards
–schremmer

Developmental Mathematics Revival!

Bringing a new vitality to college mathematics 2011 to 2019