Summer means vacations. Our two boys have always looked forward to the big trip to “the lake” in July and the smaller long weekends where we “get to” stay at motels. We eat breakfast out—something that never happens at home—in restaurants that specialize in breakfast.
One summer when the boys were small, we went into such an establishment, and our youngest wanted to buy a copy of the then relatively new USA Today. When we sat down with it, he dove directly into the sports section. Even as a six-year-old, he had an intense interest in sports, and he explained that this section was “awesome.” As we ordered and began to eat, he focused mainly on the paper. At one point, he surfaced to ask “Dad, what are batting averages?”
My answer, which I carefully tailored to fit his age and station in life, didn’t satisfy his curiosity, so he followed up with another question: How do you figure them? That question presented a challenge. At six years old, he had yet to encounter division, much less decimals and percentages. We began a colloquy on what they mean and how they are calculated, handling those concepts in context. Finally, breakfast was over. He seemed satisfied, but the rest of the trip was liberally peppered with questions about the ins and outs of batting averages.
After that I pretty much forgot about the exchange, I later learned that he had kept the subject alive. Several years hence, he came home from school with the shocking news that his teacher was confused. She was attempting to teach the class about something she insisted on calling decimals and percentages. He, on the other hand, “knew” that what she was really talking about was “batting averages.”
He had been introduced to the concepts in a specific context and, due to my disruption of the math education process and its orderly presentation of mathematical concepts, was unwilling and temporarily unable to see them outside that context. We had to take on the challenging task of generalizing the concepts beyond the application—something that turned out to be at least as difficult as learning the concepts in the first place.
His initial mindset limited him to a way of seeing the concepts called reproductive thinking. In reproductive thinking, concepts are transferred from one application to another based on the similarity of the applications. This makes sense, because the success of a particular application may well translate to success with a similar application. The greater the similarity between the applications, the greater the probability of success.
But this way of transporting concepts from application to application is limited by the small steps of similarity. My son was having trouble seeing the similarity between his knowledge of batting averages and the discussion of decimals and percentages applied to other, non-baseball applications. That is why math is taught moving from concepts to applications. That order allows for a way of seeing the potential applications known as productive thinking.
In productive thinking, basic concepts are applied to a problem space by seeing the potential for solutions from the power of the concepts to deal with the problems. Whereas a reproductive thinker is limited to applications that appear similar to problems that have already been solved, the productive thinker is able to see potential solutions to applications at the conceptual level.
Take as an example Nate Silver, the statistician and writer who once was just another six-year-old with a penchant for baseball. Silver pursued his interest in numbers and statistics into a grounding in mathematical and statistical concepts and a degree in economics. His interest in baseball has been a constant and became the subject for his work. After a detour into the world of transfer pricing with KPMG, Silver returned to his love of baseball and developed a player performance ranking system. He has since extended the application of his knowledge to handicapping political races and built a career and reputation around his work in these apparently disparate areas. This is possible through his productive (as opposed to reproductive) thinking about his discipline. Silver’s deep grounding in the concepts has enabled him to think about his solution broadly without the limitation of incremental movements between applications.
The people most susceptible to the limitations of reproductive thinking are those who have been introduced to the concepts first (or only) at the application level. That is a trap very much at work in the world of systems engineering. Because of its birth and development in the aerospace and defense market space, many (if not most) practitioners who have come to their knowledge of the discipline in that context experience difficulty in transferring the concepts to other areas unless they can find some similarity in the applications.
This is limiting in a number of ways. One of the most obvious is that our thinking is confined to only those applications that we see as similar. But the problem is more subtle than that. We adopt language and processes that come from applications with which we are familiar. These enable us to converse easily in that market space but become a barrier to those whose world is not configured in that way. We use notations, phrases, and acronyms that fail to communicate with those who live and work beyond the world of defense and aeronautics. We even advocate the adoption of standards that limit the form of our communication to specialized notations like those of the software world at the expense of communicating with those outside the specialty.
What is lacking is the conceptual view and understanding. Once the applied view is introduced, it becomes difficult to move from our specific applications to generalized concepts. That is why math is taught from the concepts to the applications and not the other way around.
As systems engineers, we need to refocus at the conceptual level. The application of systems engineering concepts is potentially quite broad. It isn’t limited to just the production of physical products or software. Human processes and public policy are ripe for systems level interventions that systems engineering can provide. But first we must see the conceptual possibilities, and we need to communicate with others in a language they can understand.