Templates in action.
Serow, Penelope "Pep" ; Inglis, Michaela
Introduction
Units of work that introduce students to circle geometry theorems
are frequently described as a string of tedious constructions. Whilst
constructivist approaches are preferred, we are sometimes forced to take
shortcuts in an attempt to meet program deadlines. To add to the
problem, when these shortcuts are taken, and a student begins following
geometrical procedures without conceptual understanding of the process,
he/she is unable to think deductively in this context.
This article presents a strategy for using Dynamic Geometry
Software (DGS) templates within a theoretical framework. Firstly, the
templates, or pre-constructed geometry sketches, enable a pre-designed
sketch to be added to, dragged to investigate properties, and facilitate
reasoning through evidence. Templates are often designed with the
support of action buttons that allow further additional information or
instructions to be viewed by the student.
Whilst many pre-constructed objects are freely available,
teacher-designed or student-designed templates better meet
students' individual needs. When designed by the teacher for a
particular class, careful attention can be given to the appropriate time
to introduce formal language and emphasis placed on particular
mathematical concepts which target the students' level of
understanding. Sinclair (2003) found that "the task question and
sketch provision must work together to create an environment for
exploration" (p. 289) and "that explicit attention to visual
interpretation and exploration using change is required in order for the
students to benefit fully from their experiences with pre-constructed
dynamic geometry sketches" (p. 313).
Secondary students of today are equipped with the skill of using
Information and Communication Technology (ICT) as a sophisticated means
of communication and a display tool. Often little flexibility is
available in various forms of mathematics education software. We
generally need to adapt our own teaching to fit the software, rather
than using ICT materials in the light of our own experiences and the
particular needs of the group of students. In the case of DGS, Jones
(2000, p. 55) identified that "students' explanations can
evolve from imprecise, 'everyday' expressions, through
reasoning that is overtly mediated by the software environment, to
mathematical explanations of the geometric situation that transcend the
particular tool being used".
Whilst it is hoped that each of the students in the class will have
passed through van Hiele's Level 2 (van Hiele, 1986), where figures
are identified in terms of properties which are seen to be independent
of one another before commencing the circle geometry unit, this is often
not the case. Some students will still be operating at this level and
the templates need to be open-ended in nature to cater for these
students. Other students will be operating at van Hiele's Level 3
where they focus on the relationships among the figures and properties.
Remaining students will be operating at van Hiele Level 4, where they
are able to focus on the complex interrelationships among properties and
figures, and hence, the place of deduction is understood. The challenge
is catering for all these students within the structure of a two week
unit of work targeting circle geometry theorems.
Van Hiele teaching phases: A teaching framework
A teaching framework that suitably addressed the need to assist
teachers in using technology to develop mathematical concepts is the
basis of the work of Dina van Hiele-Geldof. The five teaching phases
represent a framework to facilitate the cognitive development of a
student through the transition between one geometrical level of
understanding and the next. The van Hiele model acknowledges that
progress is easier for students with careful teacher guidance, the
opportunity to discuss relevant issues, and the gradual development of
more technical language.
The five phases of teaching assist in maintaining student ownership
of ideas throughout the learning process. During this process, students
can seek clarification from each other and from the teacher concerning
the language used. In particular, language plays a central role. It is
only after students have identified and described concepts, using their
own language that the more technical language is introduced. A
description of the phases is provided in Table 1 with an emphasis on the
changing role of language as the student progresses through the phases
(Serow, 2007, p. 384). The phase names have been adapted.
The five-phase teaching approach provides a structure on which to
base a program of instruction. As can be seen, the phase approach begins
with clear teacher direction involving exploration through simple tasks,
and moves to activities that require student initiative in the form of
problem solving.
Teaching sequence
The target outcome of this two-week unit of work is: "SG5.3.4
At the end of this unit, the students should be able to apply deductive
reasoning to prove circle theorems and to solve problems" (Board of
Studies, NSW, 2002).
The key ideas are: deduce chord, angle, tangent and secant
properties of circles.
There are various forms of DGS available to schools. The type
modeled in this article is Geometer's Sketchpad, known as GSP
(Jackiv, 2000). There is a range of DGS available to schools, readers
may like to explore similar activities using Geogebra and/or Cabri.
Activity 1: Information
Students work through simple constructions using Geometer's
Sketchpad (GSP) through the drawing of a circle (using GSP), labelling
and naming known features and properties, using a textbook to record
known formulae concerning the circle. Class sharing of information is
recorded.
Activity 2: Direction
Students investigate circle property relationships through the
medium of pre-constructed circle geometry templates. A series of
templates are created which when opened by the student appear similar to
Figure 1
[FIGURE 1 OMITTED]
On the revealing of information through the action buttons, and
consequent exploration and recording, the template will appear similar
to Figure 2.
Each student's constructions, measurements, explanations and
justifications are emailed to the teacher for feedback and preparation
for Activity 3 discussion. Instructions for creating templates using GSP
are contained in Appendix A.
[FIGURE 2 OMITTED]
Activity 3: New idea
Whole class discussion to formalise language conveyed in current
responses and emailed material. Class theory for each theorem is
produced based on the explorations from the previous lesson.
Activity 4: Ownership
In small groups, students sort circle geometry question cards into
theorem categories. These diagrams may be derived from typical circle
geometry textbook and exam style questions. After the students have
sorted the cards into theorem categories and have identified a
'plan' to find the unknown values, the values are identified
with justification. Class discussion follows and clarifications may be
sort using DGS. Figure 3 shows a sample of question cards.
[FIGURE 3 OMITTED]
Activity 5: Ownership
Students complete routine and non-routine circle geometry questions
from a range of sources that are not in identified categories.
Activity 6: Integration
Use GSP as a medium for the creation of individual student-designed
study guides of circle geometry theorems.
Conclusion
A main feature of the teaching sequence presented is the
integration of dynamic geometry software using the van Hiele teaching
phases as a framework (van Hiele, 1986) for maintaining student
ownership of circle geometry concepts. This is facilitated via
student-centred pre-constructed templates that acknowledge
students' individual experiences and the progression from informal
to formal language use. The teaching sequence combines a range of
effective teaching practices involving technological tools, cooperative
group tasks, whole class discussion, and a range of question types.
Appendix A: Creating templates using GSP
Making templates for individual guided learning.
Multiple pages in a document
1. Select FILE, then DOCUMENT OPTIONS.
[ILLUSTRATION OMITTED]
2. You can add a BLANK PAGE, add a DUPLICATE of another page and
give the new page any name you desire.
Hiding objects
It is a good idea never to delete parts of your diagram that are
undesirable, as many have integral functions for the diagram. Use the
HIDE function from the DISPLAY menu to hide objects that complicate the
diagram and may cause confusion for the students-- such as the
"draw" point on a circle.
[ILLUSTRATION OMITTED]
Labelling aspects of diagrams
You can label aspects of diagrams such as points, lines and angles.
Use the LABEL option from the DISPLAY menu.
[ILLUSTRATION OMITTED]
Creating a template for student use, using the HIDE/SHOW animation
tool
The following instructions are to create a Circle Geometry template
as shown in the Workshop.
1. Draw the base diagram, which the students will see on opening
the page.
[ILLUSTRATION OMITTED]
2. Type the first instruction.
[ILLUSTRATION OMITTED]
3. Select the text box using the ARROW tool, and select the
HIDE/SHOW option from the ACTION BUTTON selection on the EDIT menu.
[ILLUSTRATION OMITTED]
This gives you an ACTION icon that is preset as Hide caption.
[ILLUSTRATION OMITTED]
4. You can change the label on the action button by selecting the
button using the ARROW TOOL, and going to the DISPLAY menu and selecting
the LABEL option.
[ILLUSTRATION OMITTED]
5. To hide the text instruction you typed in point 2, just click
the action button.
6. The template can be completed by adding similar instructions and
action buttons. This is what the student will see on opening the
document.
[ILLUSTRATION OMITTED]
References
Board of Studies, NSW. (2002). Mathematics: Years 7-10 syllabus.
Sydney: Board of Studies, NSW.
Jackiw, N. (2001). The geometer's sketchpad: Dynamic Geometry
Software for exploring mathematics, version 4.0. Emeryville, CA: Key
Curriculum Press.
Jones, K. (2000). Providing a foundation for deductive reasoning:
Students' interpretation when using dynamic geometry software and
their evolving mathematical explanations. Educational Studies in
Mathematics , 44(1/2), 55-85.
Serow, P. (2007). Incorporating dynamic geometry software within a
teaching framework. In H. Reeves, K. Milton & T. Spencer (Eds),
Mathematics: Essential for learning, essential for life (Proceedings of
the 21st Biennial Conference of the Australian Association of
Mathematics Teachers Inc., pp. 382-397). Adelaide: The Australian
Association of Mathematics Teachers Inc.
Sinclair, M. P. (2003). Some implications of the results of a case
study for the design of preconstructed, dynamic geometry sketches and
accompanying materials. Educational Studies in Mathematics , 52(3),
289-317.
van Hiele, P. (1986). Structure and insight: A theory of
mathematics education. New York: Academic Press.
Penelope (Pep) Serow & Michaela Inglis
University of New England
<pserow2@une.edu.au>
Table 1. Descriptions of the van Hiele teaching phases.
Phase Description of Phase Focus
1. Information For students to become familiar with the working
domain through discussion and exploration.
Discussions take place between teacher and students
that stress the content to be used.
2. Direction For students to identify the focus of the topic
through a series of teacher-guided tasks. At this
stage, students are given the opportunity to
exchange views. Through this discussion there is a
gradual implicit introduction of more formal language.
3. New ideas For students to become conscious of the new ideas and
express these in accepted mathematical language. The
concepts now need to be made explicit using accepted
language. Care is taken to develop the technical
language with understanding through the exchange
of ideas.
4. Ownership For students to complete activities in which they are
required to find their own way in the network of
relations. The students are now familiar with the
domain and are ready to explore it. Through their
problem solving, the students' language develops
further as they begin to identify cues to assist them.
5. Integration For the students to build an overview of the material
investigated. Summaries concern the new understandings
of the concepts involved and incorporate language of
the new level. While the purpose of the instruction
is now clear to the students, it is still necessary
for the teacher to assist during this phase.
