The article that I spoke of relates to more than a single study. I think the study you a referencing is one that received a great deal of attention last week.
Another straw-man that is often set up is the one that places all non-project-based math practice into the category of mindless, manipulation of symbols. Other types of practice involve meta-cognition, categorization, and synthesis. This strengthens students' ability to "figure things out."
As time allows, I will review that research you mentioned.
John Anderson Lowell, IN
________________________________ From: John Clement <email@example.com> To: firstname.lastname@example.org Sent: Thursday, May 5, 2011 11:04 AM Subject: Re: [math-learn] Other Comments On High School Algebra 2
As I recall the article on "minimal guidance instruction" set up a straw man and proceeded to burn it. What they called minimal guided instruction does not in my experience exist. Perhaps a few misguided individuals do it, but none of the PER techniques do this. They also did not consoder any of the PER research. This article has been, in my opinion, debunked. Look at the work of Shayer & Adey or the Hellers at U. Minn. Then there is the Benezit experiment.
While complex open ended activities may inhibit some transfer to long-term memory, short practice tends to make students think that it is just a matter of memorizing without understanding. The subject of long-term memory is much more complex than just transfer of information. It is a dynamic system where recall changes existing ideas through the process or reconsolidation. When ideas are merely practiced, without cognition, the pre-existing paradigms are still there and come out to bite you.
For years people in physics have gotten students to memorize Newton's laws and gave them simple practice problems. But when asked about a real life situation which involved applying them, their "natural" paradigms overwhelmed their memorize info, and they got them wrong. I know that students have practiced invert and multiply, but still in college they will tell me that 1/s divided by s is 1. Then I ask what is 1/2 divided by 2 and they still say 1. So then I ask them to visualize 1/2 of something and divide it by 2 and they get the correct 1/4.
As to building rigor incrementally, that is a "false" idea. Rigor is often equated with the ability to do a variety of specific tasks, but in reality what the student needs to gain is enough ability to use cognition to figure out things, or be able to go to a source to find what they need to solve a problem. Then if they are to become mathemeticians they will gain high proficiency by practicing their trade.
John M. Clement Houston, TX
> > Complex problems certainly play an important role in > mathematical understanding. When those problems develop > rigor incrementally, much is to be gained. However, I > believe a balance should be struck. Educational > psychologists point to the limited ability of complex, > open-ended activities to move understanding into the > long-term memory of students. One such article, Why Minimal > Guidance During Instruction Does Not Work: An Analysis of the > Failure of Constructivist, Discovery, Problem-Based, > Experiential, and Inquiry-Based Teachingby Kirschner, > Sweller, and Clark does a good job of summarizing and > clarifying this understanding. >
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