Samuel Otten ▪ SME 842
Reflection on Gutiérrez Jaime (1998)
and Clements (1991).
In our continuing investigation of the van Hiele levels of geometric reasoning, we return to the Spanish team of Gutiérrez and Jaime. In the current article, they reflect upon some familiar work (e.g., Usiskin, 1982; Burger & Shaughnessy, 1986) and set forth to determine what type of test is best suited for determining student reasoning level and in what way the student responses on said test should be evaluated. Interestingly, Gutiérrez and Jaime on page 27 make a point that is nearly identical to one that I raised in a previous reflection connecting the research to teacher practice, namely, that interviews are the “most accurate way” to determine van Hiele levels but this is not feasible with more than a handful of students. Multiple-choice instruments, alternatively, are efficient but can be lacking in validity and reliability (see Senk, 1989). Apropos, there is what I perceived to be a direct shot taken at Senk when the authors write “The items in a test answered by a student do not define his/her level of reasoning, but the kind of answers to such items do. An item cannot be pre-assigned to a certain van Hiele level, but may be pre-assigned to a range of levels” (p 33).
Another connection that I made to previous readings and reflections occurred when I read this: “These descriptions [of the van Hiele levels] implicitly lead to the conclusion that one cannot consider a level of reasoning as a singular process that is attained (or not) by students, but must be considered as a set of processes” (p 29). At first I interpreted this to mean that the authors viewed the van Hiele levels as not a singular process and not something that is attained by the students. This led me back to my thoughts about their choice to use the word “acquired” when discussing the levels, because it would seem to contradict their rejection of the word “attained.” However, upon further reflection I feel that what the authors really meant was that the van Hiele levels are not a singular process that is attained, but a set of processes that is attained. Their “but” in the quote above rejects the singularity, not the concept of attainment by the students. This is the only way that I can reconcile this with their acceptance and pronounced used of “acquisition” terminology.
I found the most important contribution to the theory of van Hiele levels to be identification of four processes—recognition, definition, classification, proof—that exhibit different characteristics across levels. This is summed up nicely in Table 1 on page 36, and I found the descriptions of how a particular process develops across levels more meaningful than an extensive list of descriptors. Additionally, there was one theoretical point that struck me and that is the authors’ resistance to the notion of Level 5 (p 28). Were they saying that Level 5 “has not been clearly established” among middle and high school students, or among math students in general? Because I firmly believe that Level 5 does exist as evidence by many undergraduates who analyze, compare and switch between various geometric systems.
Also, remaining in a theoretical frame of mind, I felt that the authors presented a beautiful response to Ed’s concerns in class about the presence of non-hierarchical students, that is, students who reason successfully at a certain level before mastering a lower level (p 44). It was an interesting connection between this phenomenon and the misalignment of the levels between student and teacher.
Clements’s work (1991) begins immediately by presenting an alternative perspective to the conceptualization of the van Hiele levels as discrete, ordered-in-time ways of geometric thinking. Instead, they argue for “multiple paths to the development of each of multiple types of knowledge (ways of thinking) about geometric figures” (p 2). The author discussed the dynamics at play as a student works toward a balance between imagistic knowledge (using prototypes and exemplars) and declarative knowledge (consisting of known definitions, chains of reasoning, etc.). I found this section to be interesting, especially as it coalesced into the hypotheses concerning performance on various geometry tasks. This provided a new framework with which to reinterpret past readings. For instance, one could go back to the Chicago Project multiple-choice items and reflect on the performance of students who rely on imagistic knowledge versus those who rely on declarative knowledge.
I felt that Clements made good points about the presence of imagistic and declarative knowledge at all levels, though to varying degrees, but I felt it was somewhat of a mis-characterization (at least based on my limited readings) to imply that previous research conceptualized the geometric progression to be “strictly visual, then strictly descriptive/analytic, then logical, and so on” (p 5). Was there actually research prior to 1991 that argued for such a clean separation? In any event, Clements drives home the point by labeling Level 1 the syncretic level instead of the visual level.
(On page 9 there is a distinction made between “levels” and “stages.” It is not precisely clear in my mind what this distinction is, so I would like to reflect upon and possibly discuss this with classmates in the future.)
As students develop in their geometric reasoning, it is an obvious but important point made in this article that they do not lose or replace previous levels. This harkens back to the work of Gutiérrez where students were seen to revert back to previous levels when confronted with difficult or unfamiliar tasks. In my mind, the dual to this concept is that students can develop a level before they have completely mastered its predecessor.
In the interest of brevity I will condense further comments to the following: First, I do not know much about Logo geometry but am now interested. Second, I agree with Clements that distinct paths exist from the base levels to higher levels, and his distinction between imagistic and declarative knowledge seems a useful one in this regard.
References
Gutiérrez, A., Jaime, A. (1998). On the assessment of the van Hiele levels of reasoning. Focus on Learning Problems in Mathematics, 20, 27-46.
Clements, D. H. (1991). Elaborations on the levels of geometric thinking, III International Symposium for Research in Mathematics Education. Valencia, Spain.