core knowledge that all students must master to be able to develop experti=
se as a mathematician. I personally find that most drilling routines and ta=
rgets for math facts miss the point of developing appropriate context in wh=
ich to ply these skills. However, they do help to create proficiencies that=
allow students to reallocate limited cognitive resources to other areas of=
problem solving. Drawing from the work of Chase, Charness, Posner, Ericsso=
n and Anderson in the arena of developing expertise, several things come to=
mind:
1. The development of expertise requires the development of basic skills ov=
er time. There is no substitute for this core knowledge. However, this skil=
l acquisition must be context based.
2. Development over time- a lot of time. They have found that the developme=
nt of expertise takes over 10,000 hours of work/practice. However, they add=
a qualifier here- Deliberate Practice. This is then defined as systematica=
lly evaluating weaknesses, working with a coach to develop plans to reduce =
these weaknesses then systematically practicing so that stronger cognitive =
connections are formed.
3. The difference between an expert and a novice or intermediate abilities =
in a domain is the expert's ability to immediately recognize patterns. This=
does not come from simple drills that are out of context. Rather this is d=
eveloped through deliberate practice in "game", problem solving situations.=
However, the drills may allow for more cognitive resources to be applied t=
o the more abstract areas of problem solving as students have largely autom=
ated their "math fact" processes.
Much of this work was done with grand master chess players. In these studie=
s, they found that the GM players had a library of board positions that the=
y could immediately draw from when they looked at the board. However, when =
the board was arranged with pieces located randomly, their library of posit=
ions and plays was of little use and the performed only slightly better tha=
n their weaker opponents. This provides powerful evidence that context is a=
key component to skill acquisition leading toward the development of exper=
tise. These studies have now been replicated in multiple domains with simil=
ar results. When I think of most math fact speed drills, I think of a page =
with 100 de-contextualized problems placed on the page with a simple goal o=
f speed and accuracy. This will obviously help students develop their abili=
ties to do these problems with minimal cognitive effort leaving resources a=
vailable for other functions. However, I suspect that students would not pe=
rform nearly as well if these same "math facts" were embedded in contextual=
ized problems requiring them to solve real world, relevant problems. It is =
in this context that meaningful relationships and connections can be create=
d through the process of deliberate practice that can lead to the acquisiti=
on of skills and developing expertise. Therefore, while there may be room f=
or both approaches, the development of expertise is largely based in contex=
tualized practice where meaningful patterns can be studied and stored for l=
ater use.
For more in this area, I would look at the work of Chase, WG., Anderson, JR=
. Ericsson, KA. and Posner, MI.
Chris Bigenho
Director of Educational Technology
Greenhill School
4141 Spring Valley Road
Addison, TX=A0 75001
Ph. 972-628-5479
Fx. 972-628-5279
bigenhoc@greenhill.org
www.greenhill.org
=A0
Blog: http://bigenhoc.wordpress.com/
Twitter: bigenhoc
AIM: chris bigenho
Yahoo: chris_bigenho
Skype: chris_bigenho
Tapped In: ChrisWB
Delicious: http://delicious.com/bigenhoc
Diigo:bigenhoc
>>=20
>>=20
>> [ For info on ISED-L see
>https://www.gds.org/podium/default.aspx?t=3D128874 ]
>> Submissions to ISED-L are released under a creative commons,
>attribution, non-commercial, share-alike license.
>> RSS Feed, http://listserv.syr.edu/scripts/wa.exe?RSS&L=3DISED-L
>
>[ For info on ISED-L see https://www.gds.org/podium/default.aspx?t=3D12887=
4
>]
>Submissions to ISED-L are released under a creative commons, attribution,
>non-commercial, share-alike license.
>RSS Feed, http://listserv.syr.edu/scripts/wa.exe?RSS&L=3DISED-L
>
[ For info on ISED-L see https://www.gds.org/podium/default.aspx?t=3D128874=
]
Submissions to ISED-L are released under a creative commons, attribution, n=
on-commercial, share-alike license.
RSS Feed, http://listserv.syr.edu/scripts/wa.exe?RSS&L=3DISED-L
[ For info on ISED-L see https://www.gds.org/podium/default.aspx?t=3D128874 ]
Submissions to ISED-L are released under a creative commons, attribution, non-commercial, share-alike license.
RSS Feed, http://listserv.syr.edu/scripts/wa.exe?RSS&L=3DISED-L
