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Re: Benchmarks
Posted:
Mar 28, 1995 1:30 PM


>>Maybe it depends on how the counting is done. Does the child actually >>count something (blocks, beans, etc.) while doing the oral counting? Being >>able to orally count by 2s to twenty could be rote, but might be considered >>less rote if the child has to move two items for each verbalization. I >>believe that rote counting could be considered a benchmark for this grade >>level IF done in the context of a meaningful situation. This also adheres >>to the Communication Standard which talks about "opportunities to >>communicate so that the students can model situations using oral, written, >>concrete, pictoral, graphical, and algebraic methods," and "appreciating >>the value of mathematical notation and its role in the development of >>mathematical ideas," (p. 78). I would say there was a role of numerical >>language development in rote counting as well. >> >>Dawn Hoyt Kidd > > >There's a really serious issue here, and that's the question of whether >moving away from concrete representations necessarily represents a loss of >understanding. > >No no and no! We use physical representations help to *develop* >understandings, and to help *check* understandings, but the understandings >themselves can't be wedded to the physical representations. What is a >meaningful situation? Sometimes it is purely mathematical, even on the >primary level. For example (on the primary level)  what's half of 5? >Some kids will say "you can't do that, you can't have half of 5." They are >making a statement about whole numbers. Other kids will say "2 and a >half." They are making a statement about fractions. One kid is working in >one system, another kid is working in another system. Both are expressing >important mathematical understandings in an abstract context. Saying that >this discussion is only valid if they use cookies (which break in half) or >unifix cubes (which don't) is too limiting. > >The interplay between physical and quasiphysical representations and >mathematical understandings is very complex. > > > ==================================== > Judy Roitman, Mathematics Department > Univ. of Kansas, Lawrence, KS 66049 > roitman@math.ukans.edu > ===================================== > I agree with the above information. However, I am concerned 1.) that at the 2nd grade level children still have opportunities to have physical representations in learning and assessment situations, 2.) that children may be assessed in a different way than they were taught.
A meaningful level depends on the child. "Half of five" may not be meaningful to the child if s/he cannot visualize the concept. Using cookies may encourage the understanding of "half," and later other situations (with things that do not easily divide in half, or without manipulatives, etc.) can be experienced and added to the child's repetoire of knowledge of "half." I am not saying we should never use abstract representations, we may just need to work up to them, depending on the child.
Dawn Hoyt Kidd/UTAustin



