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Copyright © 2003 jsd

1  Breadth and Depth

As Mike Edmiston said, “A person who is a mile wide and an inch deep is not an educated person. But a person who is a mile deep and an inch wide is not an educated person either.”

One can portray the desired result as follows. This is the "table with legs" diagram:

    bbbbbbbbbdbdbdbbbbbbbbbbbbbbbbbbbbdbdbbbbbbb
    bbbbbbbbbdbdbdbbbbbbbbbbbbbbbbbbbbdbdbbbbbbb
             ddddd                    ddd
             ddddd                    ddd
             ddddd                    ddd
             ddddd                    ddd
             ddddd                    ddd
             ddddd                    ddd
             ddddd                    ddd
             ddddd                    ddd
Figure 1: Breadth and Depth: Table with Legs

where breadth is indicated by "b" and depth is indicated by "d". The table-top represents breadth, covering many topics, while the legs represent depth in a few selected topics. Similar ideas and a similar figure appear in reference 1. A general discussion of “thinking skills” can be found in reference 2.

Another way of saying it is “Jack of all trades and master of one or two”.

It is horribly common nowadays for people to get a pseudo-education that is broad without without being deep anywhere (“chopping the legs off the table”). This is a problem.   Having depth without breadth is a problem, too, but it is less common and usually less serious.

A student who has breadth but not depth has rather little chance of spontaneously developing depth anywhere.

It is really, really important for students to go through the all the stages of a deep investigation at least a couple of times, so they can see what the stages are. This process develops many important skills.

  A student who has depth in one area, plus some natural inquisitiveness, can probably acquire depth in other areas over time. Of course not all students have the same amount of natural inquisitiveness, so it is necessary to cover a certain amount of breadth in class. This is helpful for the less-inquisitive ones, and harmless for the rest.

This approach can be implemented by adopting one or two “themes” for the course. Each theme is investigated in depth. Along the way, there will be many opportunities to mention (but not really pursue) things in other fields that are related to the theme-field, thereby fostering some breadth.

2  Teaching to the Test

Tests are important, for a number of reasons.

For one thing, it is common for teachers to “teach to the test”. This can be a good thing or a bad thing, depending on the test.

Alas, most of the current standardized tests tend to promote breadth without depth. This is somewhat understandable, because it is hard to design a standardized test that checks for depth. If you have a group of people with disparate backgrounds, those that have depth are unlikely to all have depth in the same area.

I’ve seen tests that have two parts: the first part tests for breadth, while the second part asks the testee to choose one out of N areas and answer in-depth questions in that area. We need more tests like this. Obviously it takes extra work to create them and to score them.

It is easy to make a test or a syllabus “more comprehensive” by increasing its breadth. (If people are teaching to the test, the test is the de-facto syllabus anyway.) The obvious problem with an overly-broad syllabus is that depth suffers. This may be an unintended consequence, but that doesn’t make it any less real or any less harmful.

3  Interdisciplinary Connections

It is really valuable to have depth in more than one area.

There are a lot of people who are deep in one area or another. If you are deep in multiple areas, and can see the connections between the various areas, you may be in a good position to do things that have never been done before.

Acquiring multi-area depth is not sufficient – you have to work at making the connections. By way of illustration: I greatly admired Rolf Landauer. He was renowned as an expert in many areas, including semiconductor technology (which was his job) and reversible computation (which was his hobby). The odd thing is that he did not see the connection between his job and his hobby. It fell to somebody else (me) who knew ten times less about semiconductors and ten times less about the physics of computation to see the connection, leading to the design and fabrication of the first more-or-less reversible integrated circuits.

My professional career has almost entirely been a succession of interdisciplinary projects.

4  References

1.
Lawrence D. Woolf,
“It’s A Colorful Life”
http://www.sci-ed-ga.org/pdfs/apsclrpres042501.pdf (slide 7).

2.
John Denker,
“Learning, Remembering, and Thinking”
www.av8n.com/physics/thinking.htm
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Copyright © 2003 jsd