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Students' Reasoning about Angle Measure in Dynamic Geometry Instructional Games
Michael T. Battista, Candace Joswick, Kathryn Battista
The Ohio State University
Paper presented at the NCTM Research Conference in New Orleans, April 2004.
Introduction
Currently, both NCTM and the CCSSM focus heavily on mathematical reasoning and
sense making. In this study, we analyze the reasoning elementary and middle-school students
use, and develop, as they estimate and determine angle measure in the context of several
dynamic geometry instructional games. This study is important not only because angle is a
critical topic in geometric reasoning, but because angle measure presents an important context
for investigating the nature of the complicated link that students must construct between spatial
and numerical reasoning. As part of the larger project of which this study is part, we will in later
studies relate the detailed analysis of students' understanding of angle measure in game contexts
to their use of angle in analyzing the properties of special polygons and transformations, and
their use of number lines in some of the same contexts.
Context
This research is being conducted in the context of an NSF-funded project1 entitled
Development of a Cognition-Guided, Formative-Assessment-Intensive, Individualized ComputerBased Dynamic Geometry Learning System for Grades 3-8. The goal of the project is to create,
test, refine, and study a computer-based, individualized, interactive Dynamic Geometry (DG)
learning system for intermediate/middle school that can be used by students independently
(online or offline) or by teachers in classrooms. This learning system is entitled Individualized
Dynamic Geometry Instruction (iDGi)2. iDGi will fully integrate the use of the powerful
Dynamic Geometry instructional environment, formative-assessment, research-based learning
progressions, instructional sequencing that interactively adapts to students' learning needs, builtin student feedback and guidance, and research-based principles of educational media. iDGi will
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Support for this project is provided by Grant DRL-1119034 from the National Science Foundation. The opinions expressed,
however, are the authors and do not necessarily reflect the views of that foundation.
2 All software and screens in iDGi, Copyright 2012-2014.
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focus on 2D geometry and measurement for grades 3-8 as described in the Common Core State
Standards for Mathematics and the NCTM Principles and Standards for School Mathematics.
Theoretical Framework
According to a "psychological constructivist" view of how students learn mathematics
with understanding, the way students construct, interpret, think about, and make sense of
mathematical ideas is determined by the elements and organization of the relevant mental
structures that the students are currently using to process their mathematical worlds (Battista,
2004). To construct new knowledge and make sense of novel situations, students build on and
revise their current mental structures through the processes of action, reflection, and abstraction.
In addition to the basic learning mechanisms described above, there are several additional
mental mechanisms that are fundamental to understanding students’ geometric reasoning.
Spatial structuring is the mental act of constructing an organization or form for an object or set
of objects. Mental models are nonverbal recall-of-experience-like mental versions of situations
whose structure is isomorphic to the perceived structure of the situations they represent (Battista,
1994; Greeno, 1991; Johnson-Laird, 1983). They consist of integrated sets of abstractions that
are activated to interpret and reason about situations that one is dealing with in action or thought.
A scheme is an organized sequence of actions or operations that has been abstracted from
experience and can be applied in response to similar circumstances. In essence, a scheme
consists of a mechanism for recognizing a situation, a mental model that is activated to interpret
actions within the situation, and a set of expectations (usually embedded in the behavior of the
model) about the possible results of those actions.
An additional mental mechanism that is the major focus of our research is what we call a
spatial-numerical linked operation (SNLO) which is the connected, often simultaneous, pairing
of a spatial action/operation and a numeric operation. Examples include skip-counting by 10
while spatially marking or gesturing what are considered to be movements of 10 spatial units
such as unit angle rotations or unit lengths. Another example is spatially dividing the space
between the sides of an angle of known measure while dividing the measure of the angle in half.
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Methods and Data
In creating and field-testing an NSF-supported, learning-progression-based computer
geometry curriculum for elementary and middle school, we conducted one-on-one teaching
experiments with 19 students interacting with dynamic geometry computer modules. We spent
9-10 one-hour sessions with each student. Each student worked on a computer while sitting with
a researcher who asked the student to think-aloud while working, and who asked questions about
students' thinking. We video recorded every move the student made on the computer screen,
everything they pointed to with their fingers, and all oral discussion in a linked way.
Several times, each student played various angle games in which they estimated and
determined angle measures. The games provide various visual help for estimating and
determining angle measures and give various types of feedback on student answers, always
allowing students to repeatedly adjust their angle determinations until their answers are correct.
In this report, we focus on one game, Crazy Golf, that students played enthusiastically. To "putt"
the ball, students enter a length and angle; when they click the putt button, the ball travels to the
right the entered length then curves around counterclockwise as it sweeps out the entered angle
(see below). In Version 1 of the game, angle markers appear for multiples of 15°; in Version 2,
they appear for multiples of 30° (see next section).
We are analyzing students' work for all the angle games to determine the types of
reasoning they used to determine angle measures and how their reasoning developed and became
more refined. We are also charting each angle determination, and its amount of error, showing
moment-to-moment and day-to-day changes.
Example Results
Although our analysis is incomplete, several example case episodes illustrate part of our
analysis.
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Case 1 [KW, failed HS geometry]
Sometimes students made errors in reasoning, as exemplified below.
Target Angle 155°
KW: That's wrong [pointing at her entered angle]. Oh, I was thinking it was less on this side
[pointing to the 150° ray and the hole ray]. [Enters 155° for the angle.]
So KW had difficulty coordinating the angle measure marker and the direction of
increasing angle measure.
Case 2 [KS, going into 4th grade, had used the iDGi instructional module on angles]
Beginning Crazy Golf, Hole 1
In the first problem that KS completed in Beginning Crazy Golf, the hole angle was 135°
and hole distance approximately 215 pixels. KS first estimated 600 pixels and 90°.
KS:
Okay, I’m going to change this [deletes 600 pixels], because if I put it out too far it’ll just go over it
[motions to where the ball is off the screen]. So I’m going to change it to probably 200. And then
I’m going to change this [angle] to 120.
Int:
What made you choose 200 and 120? [KS putts ball]
KS:
The 200 because it’ll line up right with the ball and 120, wait [ball finishes putt]. Aw. Hold on.
[changes 120° to 135°]. 135. And then.
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Int:
Why’d you change to 135?
KS:
Because it’s on the 135 line [pointing to hole and along 135 degree ray]. So I think that’d be about
right. [Putts 200 pixels, 135°.]
KS:
Yeah! Oh. It needs to go up a little bit more. I’m going to do [deletes 200, enters 205] 205. Yeah.
205. [Presses putt button and message appears saying that the ball went into the hole] Because it
just needed to go up a little bit more.
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Note that at first, KS did not seem to fully understand either the distance or angle
markings on the screen. But after trying a couple putts, she understood the context. Often,
students initially do not make sense of new inscriptions that teachers assume are automatically
comprehended.
During her second day, KS moved to Version 2, Intermediate Crazy Golf.
Target Angle 130°
Estimate 1
KS:
Here’s 120 [points to 120° ray], it may be 125, so maybe 5 lines [single degree rays] could fit in
there [motions between the 120° ray marker and the hole angle ray]. So 125. [Note. The distance
that the hole is away from the origin may affect students' angle estimates.]
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Estimate 2 [after seeing the estimated angle ray sweep from 0 to 125°]
KS:
I need to change. With the 125, you go a little bit more. So I’m going to make it 130 [enters 130°,
which is correct].
Next Target Angle 110°
Estimate 1
KS:
This one, probably right here is 100 [points to where she thinks 100° is]. Cause I think 10 lines can
fit in there. Maybe half of this is 10 more [motions in the middle of 100° and 120°] so I think 110
[enters 110].
The interviewer asks how KS determined 110°.
KS:
If I separated it, I would think this is probably 10 lines [uses a finger span from 90° to where she
thinks 100° would be]. And the other half would be 10 lines to make it even. [INT uses a finger
span from 120° and to the hole ray and asks if that is half.] It’s probably 10 too. [KS motions
from 90°] This one’s 100 [motions to where she thinks 100° is] and this one is 110 [motions to
where she thinks 110°] and then add on 10 more [points to 120°] 120°.
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KS made considerable progress in determining and understanding angle measure. She
partitioned the angle spread between 90 and 120° into three composites of 10°, iterated
counterclockwise from the 90° indicator. Each composite seemed to be explicitly visualized in
terms of individual 1° "line" indicators, which we hypothesize were the 1° rays shown in the
angle module. Interestingly, even though KS had just determined an angle of 130°, she did not
see 110° as 10° less than 120°. KS' use of individual angle markers and iteration instead of
arithmetic (120+10, 120-10) is consistent with a lower position in the learning progression for
angle measure.
Case 3 [AK, going into 6th grade]
Target Angle 45°
AK:
[After affirming with the interviewer that the hole location angles are always multiples of 5°]. So it
would be right in the middle of 60 and 30. [Pointing with the mouse cursor as shown below] 30,
35, 40, 45, 50, 55, 60 [note that she pointed to 40 where the hole is]. 45.
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Int:
I thought you went 35, 40 [pointing to the hole for 40].
AK:
When I counted 35, 40, 45, 50, 55, 60, they were too close together rather than having 30 [pointing
at 30 marker], 40, 45 [pointing at the hole], 50, 55, 60 [continuing pointing with fairly even spaced
intervals until she lands on 60°]. I just jumped right into 40.
Int:
So you thought they were too close together [AK uh huh], so how did you decide 45 instead of 40
because at first you landed on 40 about here [hole] but then decided that wasn't quite right?
AK:
Yes, because it [the hole] looked like it was right in the middle of 30 and 60 and I wanted a number
right in between 30 and 60 and so I [originally] thought it was 40 but it can't be 40 because I
counted by fives up to 60 and that [40] wouldn't be even so it [40] would probably be around here
[pointing at a space definitely to the right and down from the hole], so 45 seemed like it would be
right in the middle between 30 and 60.
AK, like KS, determined angle locations by iterating composite units forward
(counterclockwise) using gestures that pointed at the endpoint of each spatial iteration.
However, AK did not seem to visualize individual angle marks within her composites, likely
indicating a somewhat more sophisticated level of abstraction of these units. Interesting, but
unknown is how KS determined that the 40 was not "even" in her iterations between 30° and 60°.
Was it because she recognized that she had placed more iterations after the hole than before?
Of course, students used other types of reasoning. For example, some students used
essentially "guess-adjust-make-another-guess" to gradually home in on angles. Students also
used fractions. For instance, one student said that the target angle looked halfway between 30
and 60 degrees, so he thought the angle was 45°.
Discussion
In addition to helping us understand the development of students' angle measure
reasoning, the present research connects to and helps us better understand reasoning about
iterating composite units, length measurement, and coordinate systems (polar). It also helps us
untangle the complicated nature of students' coordination of spatial and numerical reasoning.
Concerning the connection to length measurement, as our research on angle measure indicates,
research suggests that students construct meaningful understanding of length measurement as
they abstract and reflect on the process of iterating unit lengths and composites of unit lengths.
For instance, Joram et al. (1998) cited research indicating that, in making linear measurement
estimates, iterating a unit is the most commonly reported strategy. In one study, 81% of
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elementary students reported using unit iteration in estimation, with about one third of these
students physically marking off units, whereas the other two thirds were observed marking off
units with their eyes. Joram et al. also discussed the use of composite units in length estimation.
Comparing students' reasoning about length and angle measurement may help us develop deeper
understanding of the basic mental processes that underlie measurement reasoning.
References
Battista, M. T. (January 1994). On Greeno’s environmental/model view of conceptual domains:
A spatial/geometric perspective. Journal for Research in Mathematics Education, 25, 8694.
Battista, M.T. (2004). Applying Cognition-Based Assessment to Elementary School Students'
Development of Understanding of Area and Volume Measurement. Mathematical
Thinking and Learning, 6(2), 185-204.
Battista, M. T. (2007). The development of geometric and spatial thinking. In Lester, F. (Ed.),
Second Handbook of Research on Mathematics Teaching and Learning (pp. 843-908).
NCTM. Reston, VA: National Council of Teachers of Mathematics.
Greeno, J. G. (1991). Number sense as situated knowing in a conceptual domain. Journal for
Research in Mathematics Education, 22(3), 170-218.
Johnson-Laird, P. N. (1983). Mental models: Towards a cognitive science of language,
inference, and consciousness. Cambridge, MA.: Harvard University Press.
Joram, E., Subrahmanyam, K., & Gelman, R. (1998). Measurement estimation: Learning to map
the route from number to quantity and back. Review of Educational Research, 68(4), 413449.