Abstract
For decades, mathematics educators have been interested in engaging K-12 students in the practice of creating and using mathematical models. What might this look like in the context of geometry? Inspired by claims that students come to secondary school with knowledge of three-dimensional space that can be leveraged to engage them in modeling, we wondered what it would take to have them do so. We designed a communication task aimed at engaging teenagers in the geometric modeling of mesospace objects—three-dimensional objects of scale comparable to that of the human body. Specifically, we asked a group of teenagers to plan and enact the movement of furniture through a narrow staircase in a residential home. In this paper, we present our original design considerations, an analysis of the teens’ work, and a set of didactical variables that this analysis led us to believe need to be considered to ensure that such an activity engage teenagers in the geometric modeling of mesospace objects. The paper concludes with a discussion of the implications for research on a modeling approach to the teaching and learning of geometry.
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Notes
- 1.
We refer to our participants as teens and sometimes as youth to eschew the institutional role called forth with words like students or learners, as the first round of task design that we report here was done outside of school and without the expectation that any particular knowledge had to be learned.
- 2.
For example, a player’s management of positions in the soccer field can be quite dexterous and such practices might be describable by an observer using geometry, but that geometry may not be used reflectively by the player in enacting those practices.
- 3.
Dan Meyer’s 2010 TED talk (https://www.ted.com/talks/dan_meyer_math_curriculum_makeover) includes a similar example of a task about a water tank. In that talk, he demonstrates how the task found in the textbook could be revised in order to involve students in more of the modeling.
- 4.
Gerdes (1986, p. 12) says that “[t]here exists ‘hidden’ or ‘frozen’ mathematics” in the work of “the artisan who imitates a known production technique.” He calls it hidden or frozen because “the artisan is generally not doing mathematics” though “the artisan(s) who discovered the technique, did mathematics, developed mathematics, was (were) thinking mathematically.” Gerdes (1986) describes the work of ethnomathematicians as one of uncovering the mathematical thinking that may have produced the mathematics frozen in cultural practices such as the patterns made in basket weaving.
- 5.
The milieu, for Brousseau (1997), is the system counterpart to the cognitive agent in a task. The everyday meaning of milieu alludes to the environment. But in didactique of mathematics, milieu points specifically to those elements of the environment that inform the agent’s actions in a task (e.g., consequences of those actions, constraints on those actions).
- 6.
We also note that such assumptions about the subject are initial simplifications that address characteristics of the cognitive subject but not all of what it means to be a student. As successive design cycles take the task into institutionalized spaces like classrooms, these assumptions need to be complemented by understanding of the relationships among teacher, student, and content (i.e., the didactical contract) that envelope the cognitive subject and that are especially important in school (see Brousseau & Warfield, 1999).
- 7.
Some of the issues that we discuss are particular to the moving task, while others are issues that we expect teachers will have to struggle with whenever they create contexts for mathematical work in which they devolve to teens the choice to use the geometric theory involved in microspace conceptions of figure to understand the macrospace or the mesospace.
- 8.
The angle \( \overline{AB} \) makes with the horizontal equals the angle \( \overline{BC} \)makes with the vertical (\( \overline{BP} \)) because ∠ABC is a right angle.
- 9.
Note that the angle that the staircase makes with the floor will be congruent to ∠CBP in Figure 2b, when the length of the boxspring is parallel to the stairs. In that position, cross-section \( \overline{BP} \) will reach its maximum length among all the possible cross-sections used while the boxspring is being tilted from originally being horizontal (Figure 13.1), to an initial angle to take on the stairs (Figure 13.2a) to being parallel to the stairs (Figure 13.2b).
- 10.
Indeed, advancing up the stairs requires the movers to advance up using ∠CBP < X for an initial horizontal distance before rotating the boxspring to reach ∠CBP = X. The difference t − BP will allow for some rotations that increase ∠CBP and for some translation up with a vector that makes that angle ∠CBP with the horizontal.
- 11.
Note that we say “a task with the characteristics described above,” because we have not yet specified what objects would be moved or what space they would be moved through, even though we imagined a staircase. This was intentional: We imagine that we have assigned what would be the essential characteristics of the type of task — that the objects and space should be chosen based on the mathematics that one would like students to engage in and could expect students to engage in, if assigned the task by their teacher — and that other researchers or teachers could imagine variations of it that achieve the same goal (having students model mesospace objects).
- 12.
Having students work on tasks in pairs is common in mathematics education research, in order to make some of their thinking visible; see e.g., Lochhead and Whimbey (1987).
- 13.
Transcription: (1) “drag up the stairs to make sure not to hit the roof”; (2) “on the landing make sure top corner is positioned like so…”; (3) 2D top view of staircase with line segment, representing the conversion top, spanning the landing, with the “top corner” and “lower corner” labeled; (4) “straighten at the table and drag it up to top;” (5) “bring it up and bring it into kitchen to straighten out;” (6) “lean it [illegible] of to the right of the bathroom; (7) “try to make it stay upright” (numbers added by authors for reference, corresponding with the bullets used by the planner).
- 14.
Note that the black rectangle in Figure 13.4 represents the projection on the xy-plane (floor) of the conversion top; its side projection (say on the xz-plane in the left-most image in Figure 13.4) would show a rectangle with sides parallel to the ceiling, making an angle equal to the inclination of the stairs with the xy-plane. It is likely that the z-axis rotations made to negotiate the landing would also be accompanied by slight rotations on the x and y axes, and translations, however the instructions do not provide any evidence of this.
- 15.
Transcript: (1) “drag up initial flight of stairs”; (2) “when roof of stairs jumps up, angle the top of the frame up a little bit”
- 16.
Why the planner might have drawn the first shape as a non-rectangular parallelogram is not known to us. As he was drawing quickly, it could be that it was easier to draw vertical line segments than line segments that are perpendicular to a non-horizontal line segment (e.g., \( \overline{BC} \)), and as the angle of inclination of the object was small, the object still looked close to a rectangle. What is interesting, however, is not why the rectangle was distorted in the first model of the boxspring, but rather that the rectangle is corrected in the second model to accurately represent perpendicularity of the sides.
- 17.
Figure 13.7 represents our depiction of the measures the teens took of the couch. To ease the reader’s understanding of what the teens measured, we have labeled the extremes of the couch and use those labels A, B, C, and M, though the teens did not label any drawing they did of the couch and in fact did not draw the triangle shown in the Figure: The triangle is our reconstruction of the directions on which they trained the measuring tapes.
- 18.
Transcription: “When first person gets to landing gradually begin to increase the angle of elevation; When the second person gets to the landing straighten the couch and increase the angle until it is as straight up as possible but still not hitting the doorway at the top; When you get to the top width (?) of the stairs the couch should be completely upright.”
- 19.
We note here, even if belatedly, that the mathematics at stake in this task could be seen from at least three perspectives. From school mathematics, we could wonder what among the ideas in the high school geometry curriculum are used in and through doing this task. From the discipline of mathematics, we might add that there are aspects of doing mathematics, such as the practice of modeling, that are at stake. A third stakeholder is the ethnomathematics of household moving—which might be described as a systematic elaboration of the concepts and propositions that undergird the practice of competent movers, pretty much in the way that other ethnomathematics researchers and cultural psychologists have inspected other practices, such as carpet-laying or carpentry (e.g., Masingila, 1994; Millroy, 1991).
- 20.
This is similar to early algebra, where students may be involved in solving problems like 5 + 7 = [ ] + 3. Their work may amount to understanding operations algebraically, while students may only see explicitly actual numbers and operations among them (Carraher & Schliemann, 2007).
- 21.
We thank Mike Battista for the interesting suggestion that messages could be conveyed in video.
- 22.
We are assuming here that students may have encountered holistic descriptions of the cylinder in middle school, but they could be reintroduced to it using its definition as the locus of as a set of points in space.
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Herbst, P., Boileau, N. (2018). Geometric Modeling of Mesospace Objects: A Task, its Didactical Variables, and the Mathematics at Stake. In: Mix, K., Battista, M. (eds) Visualizing Mathematics. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-319-98767-5_13
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