Understanding also comes through practice. When the practice is carefully constructed and gains rigor incrementally, students expand the scope of the application of their mathematical understanding.
I would agree that understanding is king; however, proficiency is an effective and useful citizen.
Were you speaking to a specific curriculum or high school texts in general when you stated: "Once you understand the two common ways of solving simultaneous equations, there is not much content."
Enjoying the discussion,
John Anderson Lowell, IN
________________________________ From: John Clement <firstname.lastname@example.org> To: email@example.com Sent: Thursday, May 5, 2011 10:46 AM Subject: Re: [math-learn] Other Comments On High School Algebra 2
But proficiency comes from understanding and then practice by usage. Just getting automaticity is mindless manipulation.
The studies of successful adults merely show what they have, and not the optimum path for getting there. But there is abundant evidence that having students do problems which involve cognition and not just manipulation improve the ability to solve difficult problems. Looking at correlations does not show causality. The work of the Heller's at U.Mich showed that fewer more difficult problems work better.
Once you understand the two common ways of solving simultaneous equations, there is not much content. A few problems are really all that is needed to build proficiency. But the usual load of gobs of practice problems is merely building mindless manipulation proficiency. Unfortunately this is seen as being needed to get through the state tests because the students can't remember how to do it. But the root cause is that they have low thinking skills. I keep on saying that 75% of HS seniors do not have proportional reasoning ability, but nobody seems to want to tackle this problem. Combined with an inability to reason about 2 variable problems and the inability to sequency, the math courses are reduced to drill as a substitute for cognition.
Proficiency can comes from using things frequently over a long period of time. But when you stress proficiency (automaticity) you end up with an individual who can only do the types of problems that they have been trained for. In other words they are petrified experts with little ability to think outside of the box. This happens with many medical doctors. They rejected compelling evidence that bacterial cause ulcers and still did partial stomach removals for decades afterward.
Incidentally the current practice of assigning gobs of practice problems has a big deficit. The intelligent students are sometimes bored to tears because they immediately grap the technique and find the practice to be useless. They often turn off as a result. I had an example of really intelligent teaching in 4th grade. I took the reader home and read the whole thing because it had an enjoyable store. So during reading period I sat looking bored. The teacher asked why I wasn't reading and I told her I had already read it. After assuring herself that this was true, she let me sit in the back and read any of the books there. I was in hog heaven. She contributed to my love for reading.
John M. Clement Houston, TX
> > Any decent study of successful adults doing these things will > clearly show you that you need BOTH. Understanding and > proficiency. There is no easy out on this. >
[Non-text portions of this message have been removed]