Technology*Enhanced.Authentic.Professional.Development ...€¦ · Framework: The Learning For Use...
Transcript of Technology*Enhanced.Authentic.Professional.Development ...€¦ · Framework: The Learning For Use...
Technology*Enhanced.Authentic.Professional.Development.Framework.for.STEM.Educators.Using.Inquiry*Based.Activities!
Jhodi!Leong!ETEC!5330!Spring!2014!
!Title: Technology-Enhanced Authentic Professional Development Framework for STEM Educators Using
Inquiry-Based activities
Target Audience (Community): STEM Educators in K-12 classrooms
Context: Research has identified the insufficient amount of students adequately prepared for future careers in
STEM fields to be a result of a lack of authentic preparation in STEM K-12 education (Nadelson et al, 2012).
Although technology has been shown to improve STEM education through opportunities of exploration,
visualization, and authentic experiences, a lack of teacher confidence and self-efficacy integrating technology
into STEM classrooms is preventing effective, increased use by STEM educators (Wachira & Keengwe, 2011).
Professional development (PD) opportunities dedicated to the growth of technological, pedagogical, content
knowledge (TPCK) and the development of technology-enhanced context-specific activities has been identified
to improve teacher confidence and self-efficacy, increasing the implementation of technologies into STEM
classrooms. Based on several key characteristics identified by research of effective PD to meet teacher-reported
needs, this PD opportunity will be based on the following characteristics:
Teacher Needs Characteristic of PD Opportunity
Opportunities to collaborate with other STEM
educators on the development of curriculum-specific
content
Educators will work in small cohorts on the
development of STEM activities focused on a specific
concept for a desired curriculum
Opportunities to collaborate with working
professionals in STEM fields
The development of a partnership with STEM
professionals employed at the local university will
provide access to authentic STEM experts
Development of knowledge related to specific STEM
concepts with curricular ties
STEM knowledge will be expanded through
collaborations and discussions with colleagues and
STEM professionals, designing of inquiry-based
STEM activities, participation as a learner to complete
inquiry-based STEM activities, and opportunities to
observe group presentations on STEM content
presented by experts in the field.
Exposure to available technologies relevant to STEM An online database of STEM-specific technology tools
education will be available for reference by educators.
Development of technological skills and effective
methods of technology implementation into the STEM
classroom
Educators will have access to online tutorials about
how to use specific technologies, examples of
implementation, participation in thorough analysis of
case studies to develop an understanding of effective
methods for implementation, and opportunities to use
a variety of technologies themselves in exploratory
tasks.
Development of inquiry-based pedagogy and
implementation
Educators will have access to digital resources
highlighting the educational benefits of inquiry-based
activities, rationale of inquiry-based methods, and
effective strategies for implementation, as well as
opportunities for discussion regarding its impact on
student learning and impacts for teaching practice.
Access to continued technology support and methods
of problem-solving
Educators will form a learning community, sharing
their expertise as well as learning from that of others.
Educators will support each other in the use of
technology as well as receive support from
administration, board technology support, and online
troubleshooting resources.
Continuation of PD opportunity (ie. not limited to a
one-day session)
The scope of this PD opportunity is continual,
emphasizing continual commitment to further develop
databases and provide online support. Access to
online forums and resources can be accessed at any
time.
This program will be provided mainly through an online website hosting a variety of forums, technological
tools, and resources to develop and support a community of practice amongst STEM educators. Additionally,
this program will utilize system-wide organizational days (teachers are present at school, but students are not)
as opportunities for system-wide, face-to-face meetings amongst STEM educators. This online collaboration
will foster STEM teachers’ PD by offering continual access to and support from other STEM educators and
professionals as well as a bank of resources. The focus of these collaborations will be specifically related to the
participants’ teaching assignment, providing opportunities to develop tools and resources for concrete use in the
classroom. Teachers will also have access to online learning activities to place them in the role of a student to
analyze the learning processes of students. This will help educators develop an accurate view of concept
formation from a relational view of learning in context (Confrey, 1990). Organizational days will also allow for
experts to present learning tasks to educators to encourage them to think deeply about the content and possibly
experience subject matter from a different perspective, experiencing an inquiry-based activity first hand.
Role of Educators
Educators will consist of K-
12 STEM educators
• Educators will actively participate in online discussions regarding
pedagogy, technology, literature, effective strategies, student learning,
etc.
• Educators will actively make use of the online databases containing
educational resources such as text resources, videos, tutorials, digital
links, etc. to further develop knowledge and understanding.
• Educators will contribute to online databases with educational resources.
• Educators will collaborate with other educators and STEM professionals
to design curriculum-specific learning tasks.
• Educators will implement specific activities into their classrooms and
provide feedback to the learning community to promote further
refinement.
• Educators will provide assistance in areas of expertise to other
educators.
• Educators will maintain a contributory role as an active member of the
learning community.
Role of Facilitators
Facilitators will consist of
STEM educators considered
experts in a specific STEM
field, STEM professionals,
and school board
administration
• Facilitators will oversee the overall workings of the learning community
to ensure that contributions are valuable and representative of a
technology-enhanced inquiry-based method, as well as ensure the
appropriate use of netiquette by participants.
• Facilitators will identify the strengths of community members to use
expert resources to the fullest extent in community formation and
development (Baker-Doyle & Yoon, 2011).
• Facilitators will contribute valuable resources to online databases.
• Facilitators will contribute to the design of curriculum-specific learning
tasks.
• Facilitators will provide technological and pedagogical support to
educators.
• Facilitators will actively contribute and instigate meaningful discussion.
• Facilitators will keep up-to-date on effective TPCK impacting STEM
education and share their knowledge in the online forums.
• Facilitators will provide necessary feedback to administration and the
school board to identify teachers’ needs.
Role of Technology
Technology will consist of a
variety of digital
technologies that promote
authentic experiences in
STEM fields
• Online forums will provide the necessary collaboration space for
educators.
• Educational online databases such as Google Scholar will provide
educators with a source of scholarly reference.
• Digital technologies will be used in learning tasks performed by
educators to further develop their understandings of learning and
teaching with a tool.
• Online databases and reference tools will foster distance, anytime,
anywhere learning.
• Online design allows educators to participate from wherever and
whenever they choose.
• Digital technologies will be used in the development of authentic
activities to promote deeper understanding of STEM fields. These
technologies, technical support, examples of implementation, and user
feedback will provide valuable information to educators for effective
use.
Pedagogical Goals:
1. Participating educators will develop a greater understanding of inquiry-based instruction and learning
methods in STEM education.
a. Educators will focus on developing their skills around designing an inquiry-based activity to
achieve specific learning outcomes and effective methods for implementation using the
assignment of student and teacher roles.
b. An online forum via a website hosting blogs for discussion will promote continual access to
teachers to participate in discussion. This online forum will also be able to host a database of
resources (ie. collection of readings, studies, links to other online discussions, or access to
experts in the field) accessible by educators for further reference.
2. Participating educators will acquire knowledge about specific technologies supporting STEM education
and develop the skills required for implementation.
a. Educators will have access to a database of technologies supporting STEM education and be
encouraged to participate in discussion on each technology, activities using a given technology,
educational benefits, methods of effective implementation, and opportunities to receive/give
support for troubleshooting problems.
b. Educators will also participate in smaller projects around the development of a specific
curriculum-related task integrating a specific technology to further develop a database of STEM
activities ready for use in the classroom.
c. Teachers will actively participate in discussion on the educational benefits and affordances of
technology implementation in STEM fields.
d. Educators will extend their knowledge of available technologies and have the opportunity to
explore the usage and implementation of specific technologies individually, as well as with
colleagues, fostering opportunities for the sharing of expertise amongst professionals.
3. Participating educators will further develop their knowledge and understanding of direct links between
school-based curriculum and authentic application in STEM fields.
a. Educators will have access to online connections with STEM professionals for consultation as
well as open discussion about the development of tasks to provide authentic experiences for
students.
b. Collaboration with other educators on the development of curriculum-specific, inquiry-based
activities integrating a specific piece of technology is supported through the formation of online
communities, facilitating easy access to collaboration and shared knowledge of technologies.
c. The specific technologies to be used will be applicable to STEM fields of education, geared
towards providing authentic experiences to students by providing links to real-life applications.
Technology to be Used: The structure of this PD opportunity will be the participation in an online learning
community. A website or online learning management system, preferably hosted by the school board, will
provide an online forum for collaboration. This online forum will need to include online blogs for discussion,
allowance for uploading content, online databases of resources, and the ability to link to external online
resources.
Open databases of scholarly reference such as Google Scholar will be used to provide access to scholarly
references in the development of a collection of resources, readings, case studies, etc. that can be accessed by
educators to further develop their knowledge of pedagogy, STEM fields, implementation of technologies and
inquiry-based instruction, etc. Educators can contribute resources as well as participate in discussions about
these resources through the online forum. This will help build upon current knowledge and understandings of
pedagogy, teaching and learning strategies, previous research identifying key characteristics of effective efforts
at the implementation of technology in STEM fields, and effective strategies for implementation. Ultimately,
this knowledge will help build valuable educators that can contribute effectively to the online learning
community.
Additionally, resources pertaining to the identification of specific technologies for implementation in STEM
fields to effectively promote understanding in learners will provide educators with a variety of technologies to
instigate designs of technology-enhanced STEM learning activities. This will help educate teachers on
available technologies as well as provide examples of usage and processes in effective implementation as well
as strategies that were not effective and need refining.
Technology Integration Plan:
Goals:
1. Provide STEM educators with an online forum for collaboration and discussion.
2. Provide an online space for collaboration on curriculum-specific projects.
3. Foster online connections with STEM professionals.
4. Provide online resources to technologies relevant to STEM education.
5. Provide a forum for discussion about inquiry-based educational practices and
resources for additional reference.
6. Use expertise present in the community of practice amongst teachers and
professionals to provide a forum for problem-solving and technological assistance.
Format:
1. Online website: An online website will host a variety of forums, technological tools,
and databases of resources to develop and support a community of practice amongst
STEM educators. This will provide educators with forums for discussion,
collaboration, and support.
2. Face-to-Face Collaboration Opportunities and Learning Experiences: This
program will utilize system-wide organizational days (teachers are present at school,
but students are not) as opportunities for system-wide, face-to-face meetings amongst
STEM educators.
Framework:
The Learning For Use framework as presented by Daniel Edelson (2001) will be used to
inform the design of this PD experience as its four main principles:
1. Learning takes place through the construction and modification of knowledge
structures.
2. Knowledge construction is a goal-oriented process that is guided by a combination of
conscious and unconscious understanding goals.
3. The circumstances in which knowledge is constructed and subsequently used
determine its accessibility for future use.
4. Knowledge must be constructed in a form that supports use before it can be applied.
align with methods and processes for addressing the needs stated by STEM teachers for PD.
(See Appendix for rationale).
Step of LfU
Framework
Description of LfU Step
(Edelson, 2001)
Description of LfU Applied to PD
Experience
Motivation:
Experiencing
the Need for
New
Knowledge
Recognizing a limitation in
one’s knowledge creates a
desire to acquire new
knowledge to address an
issue or gap or to extend
one’s thinking to further
develop current
understanding. This
recognition fosters the
development of a context
for which the development
of new knowledge was
pursued, creating a memory
in one’s mind and
understanding for why the
new knowledge is relevant
and understood.
Activities must create a
demand for knowledge and
create a desire in the
learner to obtain new
knowledge. This can be
promoted through the
presentation of a task that
exposes a limitation of
one’s current knowledge
that requires new
knowledge in order to be
Experience Demand: The recognition of
insufficient numbers of students pursuing
careers in STEM fields and an inadequate
level of operational skill learned in STEM
classrooms from K-12 will motivate
educators to seek new instructional
approaches to better meet the needs of their
students, more adequately preparing them
for careers in STEM fields. The recognition
of potentially lacking skills and TPCK to
implement technology into STEM education
to further develop understanding, leading to
low confidence levels and technology
avoidance will motivate educators to seek
meaningful PD opportunities such as this to
further develop their skills and knowledge.
Additionally, more technologically
confident educators may desire greater
technology integration and a forum for
collaboration with other educators to
develop meaningful tasks. Desire for PD
opportunities to collaborate with other
STEM educators on designing curriculum-
specific learning tasks and access to
educational resources will be seen as a
valuable resource, leading to the further
development of one’s teaching practice.
Experience Curiosity: Being presented
explained. with the possibility of alternative
instructional methods such as inquiry-based
activities and instructional methods will
identify potential areas of development in
teaching practice. The identification of
these methods as beneficial to student
learning and understanding will instigate
interest about instructional approaches and
learning benefits. Educators will see this PD
opportunity as a chance to further develop
their knowledge in this area.
Knowledge
Construction:
Building
New
Knowledge
Structures
During this step, new
knowledge structures are
developed through
experiences that allow for
the addition of new
concepts to memory, divide
existing concepts into
subsections for further
analysis and development,
or create new connections
between new and/or
existing concepts. This can
be accomplished through
hands-on activities, direct
experiences, interaction
with or communication
from others, or a
combination of these
processes.
Activities must present
learners with opportunities
for direct interaction with
Observe: Educators will have opportunities
to experience inquiry-based activities from
the perspective of a learner as well as an
educator. Educators will also be provided
with opportunities to experiment with novel
technologies and/or novel strategies for
technology implementation. Experiencing
STEM-related activities and technology
integration, educators will further explore
STEM content and analyze various aspects
of relationships to real-life connections,
further developing their understandings of
the content and knowledge of the process
that learners go through in creating
knowledge structures related to the specific
content. Educators will also experience
applications of these strategies and
technologies, deepening their understanding
of the design of these tasks and effective
strategies for implementation.
Receive Communication: Educators will
have several opportunities for direct and
novel ideas and tasks that
present opportunities for
detecting relationships that
can be reinforced through
communication with others
to verify understandings
and build new knowledge
structures.
indirect communication to further develop
their understanding and knowledge. Direct
communication will come from
collaborations and discussions with other
STEM educators and working professionals.
This communication will happen at several
stages- the creation of learning activities,
and discussion of pedagogy, best practices,
effective strategies, etc., as well as
discussions about their experiences
completing inquiry-based tasks and their
assessment of the process for learners and
impact of teaching and learning. Indirect
communication will come from the
effectiveness of the learning activity in
creating meaningful learning in students and
through observations of engagement during
the implementation of a learning activity.
Knowledge
Refinement:
Organizing
and
Connecting
Knowledge
Structures
This step provides the
learner with an opportunity
to reorganize their
understanding, develop
meaningful connections
between new knowledge
and prior knowledge, and
promotes future recognition
by situating the learning
experience in a relevant
context. The acquisition of
knowledge must be situated
within a context of
application, utilizing
previous knowledge in the
formation of new
Apply: Educators will have the opportunity
to implement the activities they create in
collaboration with colleagues in their own
STEM classrooms. This will provide
educators with an opportunity to assess the
fundamental components of designing
technology-enhanced inquiry-based
activities and an opportunity to observe the
effectiveness of their ability to implement
these learned strategies effectively. Through
direct observation, educators will receive
valuable feedback about their
understandings of effective strategies for
implementation and the effect of different
strategies and design on student learning.
knowledge to strengthen
connections and foster
future recollection and use.
An opportunity for
reflection helps the learner
reinforce their
understanding and
promotes the development
of useful knowledge.
Activities must provide
opportunities for the
relevant application of
knowledge to reinforce
understanding and promote
recollections. In
conjunction with an
opportunity for reflection,
learners organize new
knowledge structures in
meaningful ways and are
encouraged to consider the
process of learning and the
links formed with prior
knowledge and
understanding. The
reflection process can help
refine understanding and
appropriately organize new
knowledge amongst prior
knowledge.
Reflect: Educators will have an opportunity
to individually reflect by assessing the
effectiveness of an implemented learning
task to achieve desired outcomes and
learning in their own classroom.
Additionally, Educators can reflect using
direct observations as well as
feedback/discussion with students,
colleagues, and professionals post-task.
This will provide educators with an
opportunity to assess and reflect upon their
level of understanding and knowledge
regarding task design, their role as a teacher,
effective strategies to elicit desired learning
outcomes for students, and the identification
of potential areas for further development.
Post-task, educators will be given an
opportunity to continue collaborations with
colleagues on the same task, using their
reflections to guide further developments
and the improvement of implementation
strategies, tasks design, or any other areas
identified for further development. Further
explorations and research into effective
strategies and developing a deeper
understanding of design, pedagogy,
authentic application, etc. will be
encouraged through access to additional
resources focused on identified areas of
improvement. Opportunities for alterations
and improvements will be followed with
additional opportunities for implementation
and further reflection and refinement.
Example of
Collaboration
Project for
Educators:
I have provided an example of one LfU-framed STEM activity using Geometer’s Sketchpad
(GSP) Software as an example of an activity that a group of educators could collaborate on to
create, implement, reflect, and refine.
(See Appendix for rationale).
Description of task: This activity will initiate student thinking about the meaning of degrees
and the potential need for an alternative form of angular measure. The design of this activity
focuses on the development of an understanding of the radian as a unit of angular measure
that can be described as a ratio between arc length and radius. A focus on developing a
definition and the ability to apply skills within an authentic context afford students with an
opportunity for knowledge creation grounded in real-life experience requiring strong
connections to previous knowledge. This activity should be completed in pairs to promote
collaboration and discussion.
Learning Objectives:
• Students will be able to define and understand the meaning of a radian as a ratio
between the radius of a circle and an arc length of equal measure.
• Students will articulate advantages to the use of radian measure for angles.
• Students will be able to solve for the radius of a circle given an angular measure and
are length.
• Students will be able to solve for an arc length given the radius of a circle and an
angular measure.
• Students will be able to solve for angular measure given the radius of a circle and an
arc length.
• Students will develop and use a method of converting between angular measures in
degrees and radians.
• Students will be able to solve application problems involving radian measure.
• Students will be able to solve application problems involving arc length.
Teacher Role:
• Motivation: The teacher will foster motivation, create demand, and elicit motivation
in students by providing support through encouraging students to explore the concept
of radians using the technology, providing technological support, presenting activities
as relevant to authentic experience, chunking activities and maintaining an
appropriate pace of exploration and instruction, and using questioning to elicit student
curiosity, as well as helping students identify potential misconceptions and areas of
understanding requiring further development.
• Construction: The teacher will guide students through the activity by offering
assistance using the technology, using questioning to provoke the identification of
relationships, encouraging exploration and analysis, assisting students in developing
links with prior knowledge and new relationships, and providing feedback to guide
exploration and evaluation.
• Refinement: The teacher will provide students with an opportunity for reflection and
encourage critical thinking about the relationships drawn between arc length, radius,
central angles and radians. Teachers will monitor discussion, offering input and
questions to further develop student knowledge and promote reflection. Teachers
will encourage students to share their work and collaborate to engage in discussion.
Additionally, the teacher will present students with a novel application of these
relationships to solidify their knowledge and broaden its application.
Student Role:
• Motivation: Students will critically examine their current knowledge to assess their
current understandings of angular measure. Students will actively complete activities
1 and 2, identifying present knowledge structures that are being challenged or
confirmed as well as being opened to the possibilities of limitations and a need for
new structures. Students will articulate their understandings and use the tools
provided to articulate further development needs.
• Construction: Students will actively participate in the construction of new knowledge
through completion of activities 3 and 4 using GSP to visualize the desired
relationships, combined with previous knowledge to thoroughly examine the data.
Students will complete these activities, ensuring their ability to articulate the process,
observations, relationships, conclusions, and remaining questions in collaboration
with others. Students will identify key aspects of their experience that led to the
creation of relationships.
• Refinement: Students will actively reflect on the learning process and identify the
mathematical relationships they developed. Students will participate in
collaborations with peers and the teacher to engage in discussions, providing relevant
information and receiving input from others to deepen understanding and promote
further thinking. Students will apply their new knowledge to actively try to solve a
novel problem.
Technology Support: Students will use GSP to develop an understanding of radian measure
by analyzing the relationship between the radius of a circle and arc length. Students will use
construction and measuring features of GSP to assist them in the construction of
understanding.
Step of LfU
Framework Activities & Design Strategies
Motivation:
Experiencing
the Need for
New
Knowledge
Fostering Motivation:
• Activity 1: Students will perform research using an Internet browser to
answer the questions: “Why are angles measured in degrees?”, “Why
is a full circle 360°?”, and “What is a degree?”
i. Through this research, students will start to discover that angular
measures in degrees are a rather arbitrary form of measurement
with no concrete mathematical foundation as to why there are 360
of them in a full circle. Students will start to wonder about the
mathematical validity of this unit of measure.
ii. The design of this activity is purposefully shaped to create a doubt
in students’ understanding about their current understandings of
angular measure. This activity will initiate critical thinking about
degrees as a form of angular measure and motivate students to
explore the meaning of angular measure in greater depth.
iii. Technology is used as a research engine to provide students with
information about angular measures in degrees necessary for this
activity. Students are exposed to a wealth of information that may
reveal misconceptions or areas of understanding that require further
development.
Creating Demand:
• Activity 2: Students will be presented with the following problem:
A construction company has recently hired you to build them a pulley
system to lift heavy objects. If the radius of the pulley must be 9 feet
and rotates at a rate of 60°/!"#, is it possible to construct a pulley
that will raise the heavy objects at least 8 feet off the ground in one
minute?
i. This problem presents students with an opportunity to critically
think about the relationship between the distance travelled on the
circumference of the pulley (ie. the arc length), the radius of the
pulley (ie. the radius of a circle), and the angle through which the
pulley rotates (ie. the central angle subtending the arc length).
Students must critically analyze the context of this question to
realize that the arc length will be equal to the distance that the
weight is raised off of the ground. Students will start to
hypothesize between these relationships.
ii. The design of this activity creates a demand for a relationship
between arc length, radius length, and central angle measure.
Students will start to identify that their current knowledge about
angular measures in degrees in insufficient to analyze this
relationship.
Eliciting Curiosity:
i. Activity 2 will elicit curiosity of a relationship between arc length,
radius length, and central angle measure once students determine
that their current knowledge about angular measure in degrees is
not providing sufficient knowledge to solve this problem. Students
will start to wonder about alternative relationships and potentially
alternative forms of angular measure that are possibly less arbitrary
than angular measures in degrees.
Knowledge
Construction:
Building
New
Knowledge
Structures
Fostering Knowledge Construction:
• Activity 3: Students will perform the following constructions and
calculations using GSP:
1. Construct 5 circles, C, of various sizes (use the circle feature on the
left tool bar in GSP). Perform the following steps for each circle.
2. Mark center O (Click to highlight O, the click “Display” !
“Label”).
3. Construct line OP (Click to highlight O and P, then click
“Construct” ! “Segment”).
4. Measure the radius OP in centimeters and label OP (Click to
highlight O and P, then click “Measure” ! “Length”).
5. Starting at P, create an arc length of length OP on the
circumference of the circle. Label this point Q (Click to highlight
P, then click “Construct” ! “Arc on circle”).
6. Construct line OQ (Click to highlight O and Q, then click
“Construct” ! “Segment”).
7. Measure and label arc length PQ and length OQ in centimeters
(Click to highlight O and Q, then click “Measure” ! “Length”,
then click to highlight P and Q, then click “Measure” ! “Arc
Length”)).
8. Calculate the circumference of each circle (! = 2!!) (Click to
highlight C, then click “Measure” ! “Circumference”).
9. Answer the following question in relation to each circle:
a. Approximately how many arc lengths PQ would you need to
go around the circumference of each circle?
10. Measure and label the central angles (an angle whose vertex is at
the centre of a circle and the arms are radii of the circle), ∠!"#,
for each circle (Click to highlight P, O, and Q, then click
“Measure” ! “Angle”).
11. Answer the following question in relation to each circle:
a. Compare the central angles, ∠!"#, for each circle. What do
you notice about all five?
b. If central angle ∠!"# = !"#!!"#$"%, write a definition for a
radian in relation to the arc length and radius length of a circle.
Following the exploration, students should have developed the
following conclusion:
• If central angle ∠!"# = !"#!!"#$"%:
The radian measure of an angle is a ratio that compares the
length of an arc of a circle to the radius of the circle. Ie.
!"#$%&"!!"!!"#$%!!"!!!"#!$% = !"#!!"#$%!!"#$%& .
Figure 1 is an example of one constructed circle :
Figure 1
" This activity has been purposefully designed to provide
students with an opportunity for the construction of knowledge
using a technological tool for assistance. This provides a
visual representation of a challenging mathematical concept
and allows students to have a tangible manipulative to explore.
" This activity will ensure that students develop an
understanding of a radian and form their own explanation.
• Activity 4: Students will use their knowledge about degrees and the
knowledge constructed in Activity 3 to complete the following,
developing a relationship between degrees and radians:
" Given a circle with radius r:
• One full circle (or one complete rotation) is _________° The arc length for this rotation is _________ or the
_________________ of the circle with equation
_________________
Therefore, !"#!!"#$%!!"#$%& = !!!!!!!!!!!!!!!!!! .
Therefore, 360° = _______!"#$"%&
• Half of a circle (or half of one complete rotation) is
_________° The arc length for this rotation is _________
Therefore, !"#!!"#$%!!"#$%& = !!!!!!!!!!!!!!!!!! .
Therefore, 180° = _______!"#$"%&
Therefore, 1!!"#$"% = _________!"#$!!"
Describe a strategy to convert from radians to
degrees:
Also, 1!!"#$"" = _________!"#$"%&
Describe a strategy to convert from degrees to
radians:
" This activity provides an opportunity for students to connect their
prior knowledge about degrees with their newly acquired
knowledge about radians, creating valuable links in their
understanding. Students must analyze their findings from Activity
3 in conjunction with prior knowledge about degrees and
circumference of a circle to develop conversion formulas in a way
that is connected to their understandings.
Observation: GSP as used in Activity 3 provides students with an
opportunity to use investigative tools to identify relationships between arc
length, radius length, and central angle measure through the exploration of
data. This provides students with an opportunity for the visualization of a
radian and allows students to use investigations to develop, use, and
interpret the data. Using knowledge from Activity 3 and prior knowledge
about degrees, students further develop their understanding of the
relationship between radians and degrees in Activity 4, connecting prior
knowledge to the formation of new knowledge. This meaningful
connection helps form connections in student understanding.
Communication: Students must complete this activity with a partner,
communicating about steps taken throughout the process. Students are
asked to directly communicate their findings in steps #8 and #10 in
Activity 3 and at the conclusion of Activity 4 to develop conversion
methods. Students will receive direct communication from the teacher as
well through guided questioning and guided assistance if necessary.
Students will receive indirect communication from the results of Activity 3
integrated with Activity 4; if students successfully complete the activity,
they should be able to draw generalized conclusion, but if not, they will
need to re-evaluate their findings and make adjustments.
Knowledge
Refinement:
Organizing
and
Connecting
Knowledge
Structures
Reflection: Activity 5: Students will reflect on the knowledge that they
discovered in Activities 1-4 to answer: “Why might radian angular
measure be useful?”. Students will use their knowledge that ! = !" to
solve the initial problem presented to them in Activity 2.
Activity 6: Students will collaborate with peers to compare their findings
and conclusions from activities 1-5. Students will engage in discussion
about process and the formation of their understandings as well.
• These activities provide students with an opportunity to verify and
solidify their understandings. Students will be able to identify with
processes taken by other students as well as learn potentially
different strategies and views that lead to the same conclusions.
Application: Activity 7: Students will be provided with the following
application problem requiring their newly formed knowledge involving
radian measure:
You have been hired to create a system of two gears to move a
conveyer belt. One gear must have a radius of 40 cm and the other
gear, a radius of 70 cm. If the smaller gear rotates through an angle of
300° in one minute, is it possible to construct this conveyer belt to
move at a rate of 2 m/min if the larger gear rotates at a rate of
1!!/150°? o Using GSP, create a labeled diagram representing the
above scenario to help you solve the problem.
" After completing the problem, use GSP to
label/calculate the desired radii, angles, and arc
lengths to verify your calculations.
• This activity provides students with a novel opportunity to apply
their knowledge and test their understandings. Students are
assisted in their learning and understandings through the visual
representation on GSP and the ability to verify their calculations.
This activity will help students solidify their understanding of
radians and its applications.
References
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Appendix
Rationale for
the Use of LfU
Framework:
The development of initial motivation is key to the LfU framework and can be achieved
through a variety of methods and activities centered on a larger goal within the context of an
inquiry-based task requiring the acquisition of specific knowledge. “The LfU approach
recognizes that for robust learning to occur, the learner must be motivated to learn specific
content or skills based on a recognition of the usefulness of that content beyond the learning
environment” (Edelson, 2001). The framework of this PD opportunity allows teachers to self-
identify an area of the curriculum, relevant to their current teaching assignment and of interest
to them, that they feel like they could deliver more effectively to promote deeper
understanding in their students. By placing educators in the role of a learner, the educators
empathize with the feelings, processes required to obtain specific knowledge, and desired
outcomes that frame the learning experience. This process deepens that educators’
understanding of the content as well as the process of delivery in implementing an inquiry-
based activity by “ground[ing] abstract understanding in concrete experience” (Edelcon,
2001). Participation in this LfU framed PD opportunity will help educators develop their
TPCK surrounding the development of their own inquiry-based activities for students designed
around the LfU framework.
The LfU framework promotes the use of technology to support discovery, stating that
“[s]cientific practice increasingly relies on computers to assist with data collection and
analysis, modeling, and prediction” (Edelson, 2001). Therefore, it is important for PD
opportunities to present educators with novel technologies, instigating ideas and the
development of technological implementation as well as to provide educators with an
opportunity to further develop skills using familiar technology to advance their skills and
broaden usage. The variety of presentation formats permitted by technology provides students
with effective forms of communication to further develop individual knowledge structures.
The requirement for further reflection and refinement post-knowledge construction gives
learners an opportunity to extend their knowledge structures past the context in which material
was learned, forming sufficient structures for future recollection, making it applicable to a
broader range of contexts. This helps address the challenge of “changing individual
conceptual frameworks… [by] challeng[ing] the overall conception help by many students and
teachers” by forcing them to consider the acceptability of their knowledge (Confrey, 1990).
Students are also forced to verify the legitimacy and rationality of their findings, exposing
potential falsities and misconceptions as well as solidifying the legitimate understandings and
connections to previous knowledge (Edelson, 2001).
One of the open issues with the LfU framework acknowledge by Edelson (2001) regards an
understanding around “how to initiate and maintain the changes in teachers’ and students’
practices that the LfU model and other constructivist learning approaches entail”. By
providing educators with a PD opportunity framed by the LfU framework as well as focused
on the development of content framed by the LfU framework, educators are placed in the role
of the learner within the framework as well as the role of the educator in the development of
tasks, developing a deep understanding of the requirements for implementation, motivations
behind the practice, and the design of activities to promote desired understanding.
Rationale for
the Use
Geometer’s
Sketchpad:
Geometer’s Sketchpad offers learners a variety of cognitive and social affordances such as:
1. The visualization of geometry though dynamic software such as GSP enables learners
to conceptualize mathematics through the use of technology to provide “access to new
understandings of relations, processes, and purposes” (Olive et al, 2010). Through
concrete experiences, students engage in the development of generalized
understandings in connection with their current understandings through “interactions
with ideas, relations, processes, structures and patterns viewed in new ways” (Olive et
al, 2010). This interaction and visualization provide a tangible quality to mathematics
not always present, providing a context in which mathematics learning takes place in
effective ways to build knowledge and understanding through work with a “model of
mathematics” (Sinclair & Jaciw, 2010). “Dynamic software packages can facilitate
visualization, connecting informal and formal mathematics, and develop perceptions of
mathematics as an instrument rather than an object” (Olive et al, 2010).
2. Technology affords students with an opportunity for operational understanding by
emphasizing the practice and applications of mathematics (Olive et al, 2010). GSP
offers students an opportunity to visualize, manipulate, and model mathematics in a
dynamic way to improve their understanding, especially in complex situations.
Students are presented with a visual representation of mathematical ideas, such as
graphs that they can be analyzed for relationships and begin to understand by
approximation not requiring the mastery of a specific skill. Although the mastery of
skills such as factoring are essential for successful advances in mathematics, skills such
as this can seem arbitrary to students when presented through a rote approach.
Alternative forms presented through the assistance of technology can help students
understand the concept and see the significance in relation to their prior knowledge and
the potential for building knowledge. “[T]his requires the student to understand and
make decisions about what mathematics might be useful and how it might be used”
(Olive et al, 2010).
3. GSP fosters meaningful interactions between task, teacher support, technological
environment, classroom and social culture, and mathematics by offing opportunities for
discussion regarding findings and process (Olive et al., 2010). Students have an
opportunity to share with and learn from other students’ prior knowledge to solve a
problem. Inquiry-based activities can be encouraged through hands-on activities
requiring active participation and knowledge creation rather than knowledge absorption
through rote memorization. Peers, the teacher, the technology, and the classroom
become resources to aid in the process.
4. GSP strengthens relationships between prior knowledge and new knowledge through
the construction of the basics as well as the novel (Olive et al, 2010). For example, a
process can start out by creating an isosceles triangle by its components, ie. by
recognizing that two side lengths and two angles must be equal, to use in the
construction of new knowledge such as the ambiguous case of the sine law. This helps
draw attention to links to prior knowledge and allows the learner to have a gradual
learning process with opportunities for small successes to encourage perseverance.
Additionally, GSP can help address misconceptions by preventing the formation of
impossible objects, ie. a triangle where the hypotenuse is shorter than another side of a
right triangle. The interaction of “Evocative Computational Objects characterized by
both their own computational nature and the evocative power caused by their
relationship with geometrical knowledge” promotes the understanding of geometric
figures and their construction to develop concrete referents (Mariotti, 2000).
5. The use of technologies such as GSP “encourage a closer relationship between
mathematical knowledge and mathematical practice, providing learners with
opportunities to experiment, visualize, and test emerging mathematical
understandings” (Olive et al, 2010). Students are becoming more educated about real-
life applications of math and future math-focused career options available.
Additionally, students are better prepared for these careers as a result of experimenting
with raw data or novel processes that require them to analyze and interpret the results
to make meaningful connections, much as they would need to do in an authentic
environment.
The promotion of technology in mathematics education benefits the learner through such
software as GSP providing opportunities for exploration and interpretation. The integration of
technology however, provides several affordances to educators as well, requiring the analysis
of effective practices (Olive et al, 2010); some considerations and affordances include:
1. Educators are encouraged to deepen their understandings of the applications of
mathematics, rather than a traditional focus on the teaching of strictly algebra-based
mathematical processes.
2. Educators develop a focus on the relationships between mathematical practice and
opportunities for students to experiment, visualize, test understandings, and extrapolate
and interpret findings (Olive et al, 2010).
3. Educators have an opportunity to experiment through the technology tools as well,
being placed in the role of a learner to better understand the processes in creating
knowledge and effective strategies for activity design.
4. Educators promote learning through encouraging students to take control and make
decisions about their learning and must develop effective strategies for guiding
explorations and encouraging drive (Sinclair & Jackiw, 2010).
5. Educators must effectively use their own feedback as well as the feedback provided by
technology to supplement student understanding and promote further inquiry (Olive et
al, 2010).
“The role of the teacher is fundamental, in order to direct the goal of the discussion and to
guide the evolution of personal senses towards the geometrical meaning of a construction
problem, and more generally to the theoretical perspective” (Mariotti, 2000). The teacher
must be adequately trained in the use of a technology to effectively implement it into an
inquiry-based activity to help students develop internal meanings and personal understandings
of mathematical content. Therefore, it is essential that PD opportunities allow educators to be
placed in the role of a student to better understand the learning process. While a tool such as
GSP can provide students with an opportunity for deeper understanding and offer visualization
not offered through conventional methods of mathematical process, this is only possible when
the activity is designed and administered in a way that fosters exploration and support.