Mile Wide … Mile Deep!

Actually, I wanted to say “kilometer wide … kilometer deep!” — but then some people would not get the reference.

At the recent AMATYC conference, I attended a session by Xiaoyi Ji titled Investigation of Math Teaching in the U.S. and China which I found inspiring. One of her main points to explain the large gap in ‘performance’ between Chinese and US students is the Chinese committment to depth AND breadth. You can see her presentation at http://www.amatyc.org/Events/conferences/2011Austin/proceedings/xiaoyiS75.pdf , and you can see the entire list of proceedings at http://www.amatyc.org/Events/conferences/2011Austin/proceedings.html.

Our recent drive to avoid a ‘mile wide & inch deep’ is a false dichotomy. The implication is that we can not have both depth and breadth. This is one that I think the Chinese system has right — we truly need a kilometer wide and a kilometer deep; depth without breadth results in students who know a fair amount about isolated pockets of mathematics … and I suggest that this is a self-defeating goal. We create more problems than we solve.

Breadth refers to two dimensions — one is the domains or categories of mathematics, the other the major areas in each domain. Within polynomial algebra, for example, we have some areas which receive most of our attention (simplifying, solving) while other areas are neglected (conic sections come to mind). We often see ‘functions’ and ‘modeling’ as alternatives, when both have a purpose. We often omit other basic forms (exponential, trigonometric). As a result, we create pockets of knowledge and chasms of ignorance … and wonder why our students have such fragile knowledge.

Depth refers to levels of knowledge, and we actually do not share a good understanding of what this means. Too often, we look at surface features of the questions we ask (skill, application) rather than a more sophisticated analysis. When better work is done, it is sometimes framed within Bloom’s Taxonomy which is not particularly well suited. A better framework for the depth of knowledge is the ‘five strands of mathematical proficiency’; you can see an excellent presentation (in fact, the original) in an online book at http://www.nap.edu/openbook.php?isbn=0309069955. This material was originally written for a school mathematics audience; however, I think you will find the concepts transfer to our level quite nicely.

Of course, we can not achieve ‘depth and breadth’ in one or two college mathematics classes. On the other hand, we can ensure either an inch deep or an inch wide in one course by the choices we make. Let us all contribute to both depth and breadth at every opportunity.

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