Everyone has comprehension of all subject matter on some level. While it is true that the level may be near "nil" in some cases (e.g. nuclear medicine in my case), the level of understanding can be increased once it is related to more familiar ideas. E.g., I took a statistics course a year or so ago and the professor asked (at the onset of his presentation) if we knew what a "cohort study" was. I was not able to explain the statistical process, however, I did know the word "cohort" literally referred to a group of common associates. I suggested to the professor that knowing what a "cohort" is, this statistical method might be somehow related to persons or things with a shared characteristic. He went on to explain the statistics involved. My statistical understanding was enhanced by the prior literal knowledge. I call this "parallel concept application" (PCA). My pebble of knowledge was strictly from the literal. It translated well at...
I was watching the movie, "The Fugitive" starring Harrison Ford when I heard something that gave me pause, mathematically speaking. After Richard Kimble escaped the first time by way of the bus/train accident, the small town sheriff was relieved of the case by another official (who, by the way, referred to the sheriff as "Wyatt Earp" because of how he was handling the situation ~ not nice!) The relieving officer, debriefing those who were assisting, claimed that Kimble had been running for about 90 minutes at a foot speed of about 4 miles per hour. Therefore, they had about a 6 mile radius. Was this an accurate, overstatement or understatement of the situation?
The answer to this question is, "one bite at a time". That's the approach I encourage students to use when approaching math problems. Often in reading an apparently difficult problem, looking at each part in its singular concepts, they are able to work through the problem to a successful end. They start at a place where they are comfortable and usually find the problem becomes progressively solvable. This encourages them to integrate skills they have used for years in the context of the current problem thereby expanding their knowledge base...'til the next time!