Exploring mathematical models using technology and its impact on pedagogical mathematics knowledge by Jennifer Suh George Mason University

1
Exploring mathematical models using technology
and its impact on pedagogical mathematics
knowledgeby Jennifer SuhGeorge Mason
University

• Presentation at Association for Mathematics
  Teacher Educators
• Irvine, California - January 2007

2
Developing highly qualified mathematics teachers
by

• Developing Pedagogical Content Knowledge
• subject matter knowledge,
• pedagogical knowledge, and
• knowledge of context.
• "from being able to comprehend subject matter for
  themselves, to becoming able to elucidate subject
  matter in new ways, reorganize and partition it,
  clothe it in activities and emotions, in
  metaphors and exercises, and in examples and
  demonstrations, so that it can be grasped by
  students"
• (Shulman, 1987 13).

3
Why is modeling mathematics concepts important in
developing pedagogical content knowledge ?

• PEDAGOGICAL CONTENT KNOWLEDGE is knowing the
• most useful forms of representation of these
  ideas, the most powerful analogies,
  illustrations, examples, explanations, and
  demonstrations-in a word, the ways of
  representing and formulating the subject that
  make it comprehensible to others" (Shulman, 1987,
  p. 9).

4
Mathematical Teaching Knowledge
5
Why are mathematical models so important?

• Provide real life context
• Help students build a network of mathematical
  relationships.
• Consider what ideas or models might be the most
  accessible to students)
• Plan the developmental progression of models (how
  models evolve)

6
Effective Representational Models have

• Transparency how easily can the idea be seen
  through the representation
• Efficiency Does the representation support
  efficient communication and use?
• Generality Does the representation apply to
  broad classes of objects or concepts?
• ClarityIs the representation unambiguous and
  easy to use
• Precision How close it the representation to the
  exact value?
• From National Research Council (2002) Adding it Up

7
Importance of mathematical modeling and
representations

• Although 43 of the US teachers successfully
  calculated 1 ¾ / ½ , almost all failed to come up
  with a representation of division by fractions.
• Among 23 teachers, 6 could not create a story and
  16 made up stories with misconceptions.
• From Liping Mas (1999), Knowing and Teaching
  Mathematics

8
Five important consideration for technology
integration(Garofalo, et al., 2000, p. 66)

• Introduce technology in context
• Address worthwhile mathematics with appropriate
  pedagogy
• Take advantage of technology
• Connect mathematics topics
• Incorporate multiple representations

9
Research Question

• How does activities with mathematical modeling
  and representations of virtual manipulative
  facilitate the development of pedagogical math
  content knowledge?
• What experiences can increase teachers
  competence and confidence in integrating
  technology and working with manipulative models
  in mathematics?
• How are technology applets implemented into
  lessons as concept tutorial/models, tools for
  investigations or free explorations in teaching
  and learning mathematics?

10
Methods

• Participants
• Twenty-one preservice teachers in Elementary
  Mathematics Methods classroom
• Data sources
• Teacher surveys with likert scale open ended
  questions
• Artifacts from class activities
• Technology lesson plans

11
Procedure Mathematical modeling activities

• Process 1 Relearning the mathematics content as
  a teacher.
• Process 2 Selecting and evaluating for
  appropriate mathematical models
• Process 3 Implementing the mathematics model in
  a lesson

12
PROCESS ONE Relearning the mathematics content
as a teacher

• Define the mathematical essence in a lesson
  (Knowledge maps)
• Developing relational understanding through
  lesson study

13
PROCESS ONE Relearning the mathematics content
as a teacher

• Define the mathematical essence in a lesson
  (Knowledge maps)
• Developing relational understanding through
  lesson study

14
PROCESS ONE Relearning the mathematics content
as a teacher

• Define the mathematical essence in a lesson
  (Knowledge maps)
• Developing relational understanding through
  lesson study

15
PROCESS TWO Mathematical modeling activities

• Selecting and evaluating for appropriate
  mathematical models (math model evaluations)
• What are the different uses of these models?
• How are students developmental needs supported
  by the different uses of these models?
• Is there a developmental progression for these
  models? Are some used of the model precursors to
  others?
• What role does the model play in helping students
  visualize different strategies?
• How might one represent a given strategy with
  each model?

16
Modeling perimeter and area
http//www.shodor.org/interactivate1.0/activities/perm/index.html http//nlvm.usu.edu/en/nav/frames_asid_281_g_2_t_4.html?openactivities
http//www.funbrain.com/cgi-bin/poly.cgi http//www.mathplayground.com/InteractiveGeometry.html
17
Modeling ten-ness
18
Modeling ten-ness
19
build deeper understanding through multiple
representations

• Example of Mathematical Model to teach density of
  numbers

20
from tools to represent thinking into models for
thinking
21
PROCESS THREE Implementation phase

• Implementing the mathematics model in a lesson
  (lesson design creating a task sheet to guide
  learning)

22
Preservice teachers reported that the virtual
manipulatives website were

1. New ways to model math ideas
2. Strategy for students to use for collaborative
   knowledge construction
3. User-friendly and handy visual model for visual
   learners
4. Tools to entice reluctant learners and special
   ed. students who could not focus on work with
   manipulatives

23
Technology integration survey
24
(No Transcript)
25
(No Transcript)
26
(No Transcript)
27
(No Transcript)
28
(No Transcript)
29
(No Transcript)
30
(No Transcript)
31
Path to pedagogical content knowledge

• American educators assume that you need to know
  content knowledge before you can plan lessons.
  Chinese teachers think you learn content
  knowledge by planning lessons.
• from Liping Mas (1999) Knowing and Teaching
  Elementary Mathematics

32
In its 2001 report, the National Council for
Accreditation of Teacher Education (NCATE, p. 7)
calls for an integration of technology as
summarized below

• QUESTION To what degree are higher education
  institutions meeting their responsibility for
  preparing tomorrows classroom teachers?
• ANSWER Bluntly, a majority of teacher
  preparation programs are falling far short of
  what needs to be donecolleges and universities
  are making the same mistake that was made by K-12
  schools they treat technology as a special
  addition to the teacher education curriculum
  requiring specifically prepared faculty and
  specially equipped classrooms but not a topic
  that needs to be incorporated across the entire
  education programTeachers rarely are required
  to apply technology in their courses and are
  denied role models of faculty employing
  technology in their own work.

33
Methods classes and professional development
training

• In the student approach, preservice teachers use
  the same technology their future students will be
  using. They explore concepts via the aid of
  technology, and then decide how they will use
  that same technology to explore those concepts
  with their future students.

34
Implications for teacher educators

• Within the context of their content and methods
  courses, preservice teachers utilize technologies
  they would use with their own future students
• Meaningful activities and technologies are
  selected purposefully (not just technology for
  technologys sake)
• Technology integration occurs throughout the
  teacher preparation program, not just one
  semester, for example and
• Faculty use technology as a means for modeling
  and for representing content and pedagogy