University of Wyoming
8TH Annual Math & Science Teacher’s Conference at Casper College, 2010
Description of the Program
• The virtual master’s degree program, offered
jointly by the University of Northern Colorado
and the University of Wyoming, started in the
summer of 2009 for secondary mathematics
teachers to improve mathematics
achievement in secondary education in the
Rocky Mountain region.
Vision
• Increase student understanding of
mathematics by promoting a master’s degree
program for 7-12 grade mathematics
teachers, which aims at developing highly
qualified, culturally competent, and
pedagogically effective mathematics teachers
in Colorado and Wyoming.
Why virtual?
• The Rocky Mountain region is a vast, sparsely
populated, and rural region where teachers
lack access to professional development
programs (PD).
• In-service teachers are not able to leave their
positions to attend PD.
• The number of faculty to provide PD in such a
vast region is limited.
• By means of alternative delivery formats, we
make the master’s degree accessible to
teachers in their home region.
Why important?
• Online teacher professional development
programs are gaining an important place in
higher education as a means to enhance
students’ achievement in schools
• Online professional development programs
(OPD) for teachers are also increasing as the
rapid development of technology allows
distance education to grow.
• Virtual or online degree programs provide
teachers with flexibility without the constraints
of geographical borders or time.
Goals for Math TLC
Develop a shared vision of mathematics as a culturally rich
subject in which K-12 mathematics proficiency is defined by
shared community standards
Expand mathematical content knowledge in ways that
broaden exposure to mathematical ideas and deepen
understanding of topics that extend K-12 mathematics
content
Increase pedagogical content knowledge by examination
of how students think and learn about mathematics.
Empower participants as lifelong professional learners who
regularly reflect on themselves, students, and community
context to improve teacher practice and student learning
Produce a research-based and tested model for master
teacher development based on the above goals that
improves mathematical achievement for all students.
The goals of the master’s program
Extend teachers’ content knowledge beyond the content
they customarily teach to students
Engage teachers in actively building their pedagogical
content knowledge (Shulman, 1987), enabling them to
enlarge their repertoire of pedagogical methods, skills
and knowledge congruent with standards (NCTM, 2000;
CDE, 2007; WDE, 2003).
Support teachers to evaluate and improve their
teaching practice by regularly engaging in lesson
experiments and small-scale action research projects.
Develop teachers’ knowledge, skills and disposition to
effectively teach math in a culturally diverse classroom.
Develop a shared vision of mathematics as
a culturally rich subject in which K-12
mathematics proficiency is defined by
shared community standards
Mathematics as a culturally rich subject
• Mathematics is constructed within a cultural
context.
• Stigler (1989) says, “Mathematics itself
consists not only symbolic technologies but
also of other cultural products-such as values,
beliefs, and attitudes […] Educators should
teach the values of mathematical culture, not
because it will help children learn
mathematics, but because values are part of
the mathematics that children should learn.”
(p. 369).
Mathematics as a culturally rich subject
• In ancient times, in Egypt, farmers along the
Nile river needed Geometry. The Nile would
flood the land and destroy the farm areas
each year. When the waters receded, the
boundaries had to be redefined by
measuring fields.
• The Egyptians (about 1800 BC) had
accurately determined the volume of the
frustum of a square pyramid.
MCRS (Cont’…)
• Babylonians’ (2000-1600 BC) geometry was
empirical, and limited to those properties
physically observable. Through
measurements they approximated the ratio of
the circumference of a circle to its diameter to
be 3.
• The Maya number system was a base twenty
system. One of the Maya’s calendar, Tzolkin,
composed of 260 days, 13 months of 20
days.
Develop teachers’ knowledge, skills and
disposition to effectively teach mathematics in a
culturally diverse classroom
What is Culturally Responsive
Teaching (CRT)?
Culturally Responsive Teaching (CRT)
Gay (2000) defines
culturally responsive
teaching as using the
cultural knowledge, prior
experiences, and
performance styles of
diverse students to make
learning more
appropriate and effective
for them; it teaches to
and through the strengths
of these students.
A learning experience drawn from a
legend of Maori people in New
Zealand
In the South Island [of New Zealand] there is a lake
whose waters, by day and by night, rise and fall. The
Maori people know that the pulsing of the water comes
from the beating of a giant’s heart, the heart of Matau
who was burnt by the brave Matakauri. The waters of
Lake Wakatipu rise and fall about every five minutes. If
the lake was formed about 1000 years ago, how many
times has Matau’s heart beat since then? If Matau’s heart
beat once every three minutes, how many times would
Lake Wakapitu have risen and fallen over the last 100
years? (Heays, Copson, & Mahon, 1994, p.8 as cited by
Averill, Anderson, Easton, Te Maro, Smith, and Hynds
(2009).)
Literature suggests
• Developing a knowledge base about cultural
diversity
• Learning mathematical content from ethnically
and culturally diverse origins
• Participating in and building a caring
community of learners-this includes
developing ways to calibrate teacher
intentions with student perceptions.
Literature suggests (Cont’…)
• Seeing personal communication patterns and
using that awareness to learn to
communicate effectively with diverse
students.
• Responding supportively to socio-economic,
cultural, and ethnic diversity in the delivery of
instruction.
Characteristics described by Gay
(2000)
• It acknowledges the legitimacy of the cultural heritages of different
ethnic groups, both as legacies that affect students' dispositions,
attitudes, and approaches to learning and as worthy content to be
taught in the formal curriculum.
• It builds bridges of meaningfulness between home and school
experiences as well as between academic abstractions and lived
sociocultural realities.
• It uses a wide variety of instructional strategies that are connected
to different learning styles.
• It teaches students to know and praise their own and each others'
cultural heritages.
• It incorporates multicultural information, resources, and materials in
all the subjects and skills routinely taught in schools
Some of the Principals of CRT
• Communication of High Expectations
There are consistent
messages,
both
There are from
consistent
the teacherfrom
andboth
the the
messages,
whole
teacherschool,
and thethat
whole
school,
that
students will
students
will
succeed,based
basedupon
upon
succeed,
genuine
respect
for for
students
genuine
respect
and belief in student
students
and belief
capability.
in student capability.
Principals of CRT
• Active Teaching Methods
Instruction is
designed
promote
There are to
consistent
student engagement
messages,
from both the
by requiring
that
teacher
and the
whole
school,
thatplay
students
students
an will
succeed,
based
active role
in upon
genuine
respect
for students
crafting
curriculum
and belief in student
and developing
capability.
learning activities.
Principals of CRT
• Teacher as facilitator
Within an active
teaching
There are consistent
environment,
messages,
fromthe
both the
teacher's
rolethe
is one
teacher and
whole
school,
that
students will
of guide,
mediator,
succeed,
based upon
and
knowledgeable
genuine
respect
students
consultant,
as for
well
as
and belief in student
instructor.
capability.
Principals of CRT
• Inclusion of Culturally and Linguistically
Diverse Students
There is an ongoing
participation in
dialogue with
students,
parents,
There are
consistent
and
community
messages,
from both the
teacher
and on
theissues
whole school,
members
that
students
succeed,
important
to will
them,
based
respect
alongupon
withgenuine
the
for students and belief in
inclusion of these
student capability.
individuals and
issues in classroom
curriculum and
activities.
Principals of CRT
• Cultural Sensitivity
To maximize
learning
opportunities,
teachers
gainconsistent
There are
knowledge
of theboth the
messages, from
teacher
andrepresented
the whole school,
cultures
that
students
will succeed,
in their
classrooms
based
upon genuine
and translate
this respect
for students and belief in
knowledge into
student capability.
instructional
practice.
Extend teachers’ content knowledge beyond the content
they customarily teach to students
Polygons on earth
Regular Hexagon in chemistry
Snow flakes- Regular Hexagon
Honey comb-Regular Hexagon
The name of
polygon
N= 3
N=4
N=5
N=6
The number
of diagonals
passing by
one vertex
The number
of diagonals
The number
of triangles
formed by
drawing all
diagonals
passing
through one
vertex
What is the
total sum of
the measure
of the interior
angles?
What is the
total sum of
the measure
of the
exterior
angles?
For n-sided polygon
• Find the number of diagonals passing by one
vertex
• Find the formula that determines the number of
diagonals.
• Determine the number of triangles formed by
drawing all diagonals passing through one
vertex
• What is the total sum of the measure of the
interior angles?
• What is the total sum of the measure of the
exterior angles?
Definition
• Let v0, v1, v2,...,vn where n  3 be a set of
points in the plane such that v0  vn .
• The union of the line segments v0v1, v1v2 ,...,vn1vn
is an n-sided polygon
 whose vertices are
v 0 , v1, v2 ,...,vn and whose edges are the
line segments v0v
1, v1v
2 ,...,vn1vn .


A student might ask…
• Although the definition
admit n  3 ,
what if n=1, n=2? How can it
be called and represented?

Response…
In spherical geometry,
it is possible to have a
two sided polygon,
namely digon
Spherical triangle
Euclid’s Postulates on the Sphere
• Postulate 1: Two points determine a unique
line
• Is this postulate valid in spherical geometry?
Spherical Geometry Terms
• Polar points are the end points of the sphere’s
diameter (called as antipodal)
• Straight lines are great circles; that is circles
that contain any pair of polar points.
• A segment (arc segment) on a sphere is the
shortest distance between two points.
• A segment is always part of a great circle,
and is also called geodesic.
Response to the question
• If two points are antipodal, then there are an
infinite number of straight lines containing
these two points, such as all the lines of
longitude. Euclid’s first postulate is therefore
not valid in spherical geometry.
Euclid’s parallel postulate
• Given a line L and a point P not on L, there
exists a unique line though P parallel to L.
• How would you re-word this postulate so that
it is true for spherical geometry?
• Given a line G and a point P not on G, every
line through P intersects G; that is, no line
through P is parallel to G.
Engage teachers in actively building their
pedagogical content knowledge (Shulman,
1987), enabling them to enlarge their
repertoire of pedagogical methods, skills and
knowledge congruent with standards (NCTM,
2000; CDE, 2007; WDE, 2003).
Cone Project
• Teacher participants explored geometry on
the cone in groups of three.
• Teacher participants explored what interested
them, wrote a report, and presented their
findings to their classmates.
• Engaging in such a project will enable them to
apply the same pedagogy in their practice.
Empower participants as lifelong
professional learners who regularly reflect
on themselves, students, and community
context to improve teacher practice and
student learning
Action Research Project (ARP)
ARP Example
Courses & Curricula
• The Math TLC provides 30- credit hour, 2year master’s program, which involves faceto-face and online courses.
• Of 30 credits, 18 hours mathematics and 12
hours are mathematics education courses.
• All courses address NCTM principals and
standards: teachers experience as learners
the kinds of instruction recommended for K12 students.
Mathematics Courses
Mathematics Education Courses
Research-based course design
characteristics
• Employ a research-based instructional
design, modeling learning strategies that
promote effective classroom learning and
teaching, and that teachers can also use with
their students. (Bybee, 2006)
• Build new knowledge upon teachers’ prior
knowledge.
• Support learning through interaction among
teachers about mathematical ideas
Research-based course design
characteristics (Cont’…)
• Convey clear purpose and outcomes. (NRC,
1999, 2005).
• Incorporate a variety of learning activities to
engage teachers, appeal to different learning
styles, and explore the cultural capital of
teachers and the students they teach
(Bourdieu, 1986; Civil, 2002; Kuhn, 2005;
NRC, 2000).
• Assess teacher understanding frequently.
Research-based course design
characteristics (Cont’…)
• Situate learning within meaningful, relevant
contexts (e.g., action research; Nentwig &
Waddington, 2005).
• Cultivate a safe, non-threatening, low-risk
environment for teachers to express new
ideas and try out new approaches, such as
incorporating collaborative learning strategies
within course designs.
Questions??
References
• Averill, Anderson, Easton, Maro, Smith, & Hynds (2009). Culturally
responsive teaching of mathematics: Three models from linked
studies. Journal for Research in Mathematics Education, 40(2), 157186.
• Bishop, A. J. (1988). Mathematics education in its cultural context.
Educational Studies in Mathematics, 19(2), 179-191. Retrieved from
http://www.jstor.org/stable/3482573?seq=2
• Bishop, A. (1991). Mathematical enculturation: A cultural perspective
on mathematics education. Dordrecht: Kluwer
• Ross, S. W., (2000). Non-Euclidean Geometry. Retrieved from
http://library.umaine.edu/theses/pdf/RossSW2000.pdf
• Stigler, J. W. (1989). Review: Mathematics meets culture. Journal for
Research in Mathematics Education. 20(4), 367-370. Retrieved from
http://www.jstor.org/stable/pdfplus/749442.pdf
• http://www.intime.uni.edu/multiculture/curriculum/culture/Teaching.htm
•
THANK YOU