Dick Lesh, one of 150 PhD's currently doing research for ETS, has come to the Geometry Institute at Swarthmore (see geometry.institutes for more information). He is knowledgeable about instruction, psychology, higher level mathematics and the world of technology, making him an ideal candidate to be reassessing assessment as it is currently visualized within the sphere of mathematics both by ETS and by many teachers, students, and parents. An outline of the talk he gave us this afternoon follows.
Some themes: --A single number cannot accurately characterize a student. --Mathematics is a growing field in a changing world; teaching strategies tend to view both as excrutiatingly fixed. Questions involving systems (telecommunications, marketing, etc) have become more relevant then questions involving addition of oranges. --(a related thing item) We need to pay attention to a broader range of abilities than we currently tend to do. Searching for more abilities results in finding more people with more abilities. Dick talks of how time and time again he has heard very sophisticated problem solving techniques from students who had just about been given up on. This broader range need not leave mathematics, but ought to focus on more aspects of the huge mathematical discipline that aren't ordinarily touched upon at the precollege level. --Students need to know who is asking a question and why: context is key. (Example given: "How do I get from Swarthmore to New York?" is not a straightforward question. It provokes more questions or several answers: "Is time important to you?" "Is scenery important to you?" "Is money important to you?" / to what degree?, etc. ) A suggestion for finding such questions is to have students look for newspaper articles which involve math and to have them construct questions based within the context of a given article.
Now for some more information more directly related to Assessment --activities in assessment should also be learning experiences. One can simultaneously learn and document learning. --assessment as a giving of information to decision makers: 1) who are they? 2) what decision? 3) policy makers? --assessment should not occur entirely at the end of a course: it should be going on continuously from the beginning through the middle and end of the course. --we need to get kids talking about what they are learning and then use what they say to judge where they are. --What exists and how this may change:
__Traditional testing__ __Instructional assessments__ Testing = ordeal Examining = studying closely Measuring = partitioning into equal Describing = generating a multi-dimensional fragments (units), measuring verbal portrait general intelligence Evaluating = assigning a value Assessing: taking stock, getting oriented (one dimensional and uncon- (give profiles of the sort used by ditional) businesses instead of a single number) Goals: Make tests more credible Make assessment more authentic
(It should be noted that Dick Lesh came out strongly in favor of the methods described in the second column.)
In many situations, tests are not the best means available of assessment. Reports and group projects should be described in a portfolio set up by the students, and noted on a resume. The projects should undergo some sort of official process that will document their existance similar to that undergone by the reports and projects of professionals.
Lesh especially stressed the importance of current trends in psychology, most specifically how cognitive psychology lends itself to math education. People map information onto images. Often problems arise when children (and I would extend this to anyone learning new things) have heard presented information, but don't know what to do with it, as they are unfamiliar with the framework of the ideas and models. Many young children, when asked to construct a cube, make something that looks like: .__. .__|__|__. |__|__|__| (hope this came out okay. to me it looks like a box with |__| unfolded edges. rather cubist, really)
And he showed us the following . . . . and asked us what we saw. Then he connected some dots. . . .
He showed several different patterns, among them, the following: . ./ | \. | \ | / | The first time we saw this I think most of us saw it as an asterix | / | \ | within a hexagon. After seeing a few more images, we were amazed \ | / to see a cube. ' People often need to be trained to see things in different ways. (Just as I would imagine, assuming this posts nicely, that those of you more accustomed to character graphics will have an easier time understanding the above image than those of you to whom this may be a rather new way of seeing.) Lesh feels that for learning to occur the most important and relevant models must be stressed.
We did an activity in which given a quadrilateral and a point light source (such as a candle), we had to deduce whether a square shadow could be produced. (The given shapes were: a square, an isoceles trapezoid, a kite, a rectangle, a rhombus, an irregular quadrilateral, a concave quadrilateral, and a parallelogram.) This is one of the activities which has elicited very impressive solutions from D students in math.
Lesh finished by proposing a teaching unit beginning with "model construct". This would occur during an activity like the above. (For students unused to this sort of activity, the activities don't really tend to work effectively until the third try at them. Lesh has found that it takes this long for them to stop looking to the teacher for the "right answer" or for more specific instructions as to how to proceed. ) The second part of the unit would be "model explore." This is more pure math and deals with situations to which the model can be applied. The unit finishes with "model apply" in which real world applications come up.
ETS is working on 10 such units, which teachers can adapt to fit their own time constraints. The units need not be used as a block.
Look for participants in our geometry software institute to be airing their reflections.