How to teach or “transfer training” or “get others to be able to do it and know it, too”

I just came across a passage that does a good job of conveying the essence of how to teach.  It is in ‘Higher Education in Transition” 4th Ed. by John S. Brubacher and Willis Rudy.

Since this passage essentially describes the basic philosophy that I use in teaching, and since I teach math, I will say something about how I teach, first.

In my second or third year of teaching, I was trying to explain some difficult material.  That is when I realized that students at every level of math (I will give an example, later.) need to learn to approach problems the same way that I approach a research problem and that there are some basic ideas in math that occur at every level.  (This is not a new idea.)  Furthermore, I realized that I can’t tell students “Use this idea, use this tool”.  I had to guide them into using the ideas.  So here is an example from high school algebra.

Example 1:  Suppose we want to solve “something positive squared” is equal to 4.  That is easy.  Suppose we want to solve “something plus 1 squared” is equal to 4.  Just change it back to the first problem, that is, transform it.  Now, how do we transform x^2 + 2x + 1 = 2?  We look for the tools, but we start by using the idea of transforming, etc…Then the quadratic equation is easy to derive.

Example 2: (more sophisticated, assuming more background from the reader)  Many books derive the equation of a plane and then solve problems about lines and planes.  Most students completely miss the idea when this is done.  What is the idea?  What do you mean when you say “equation of a plane”?  Actually,you mean something very deep and useful in math.  You mean that there are tools in algebra and there are tools in geometry and to be good at math one must understand how to translate problems back and forth for both understanding and computation.  Here is what works well for me.

I explain that there is algebra on one side and geometry on the other and that vectors and vector algebra can be used to make the connection.  Of course, I then have to start teaching the detailed use of the tools.  Once that is done, I write down the equation,

ax + by + cz = d

and ask what it describes.  I eventually guide them into asking what it says about vectors. (Remember we can use vectors to go from algebra to geometry. For example, ax + by=0 says <a,b> and <x,y> are perpendicular (sort of, it really means (or can mean) that the line segments from (0,0) to (a,b) and  and to (x,y) are perpendicular) I help them understand what it says.  From that we derive the equation of a plane.  We then use this “habit of thinking” along with the “basic concept” of transforming one type of problem into another type..

Now notice in the above, there is another “basic concept” and another “habit of thinking”.  It is this.  In math something either is a thing or it isn’t that thing.  There is a definition of the “thing” that is verifiable.  That is a basic concept.  Being very careful in our thinking, especially when issues get hard or subtle for us, is a habit of thinking that we need to get into, by watching others do it (and being aware that that is what they are doing) and by realizing on many occasions we can get out of a jam by just being careful.

Here is the quote from the book.

“First…to get transfer, the instructor had…to strive for it…second…[there need to be]…identical elements in the items learned and the items to be improved by the transfer…identical elements… principally of two sorts…habits of mind…[and]…basic concepts…”

This doesn’t seem to be too far off from what I described above.