Discussion 2 62104951 by 3O6K6N


									Irma Crespo                                         EDD 514                        October 2, 2009

Posted by Irma Crespo on Tuesday, September 29th, 2009 12:02 PM

Wow! There's so much information out already. I just checked yesterday and there weren't much.
I was worried about being redundant from our first discussion but I wrote something about how
mathematics came to be. I'll research more on the classroom perspective and peruse more on
the other pieces suggested for reading.

What is mathematics?
         In order to understand the obvious impact of modern technology on mathematics
education, it is worthwhile to step back in time and find out how far mathematics has gone. As
Grandgenett said, posing the question on what the study of mathematics is "is not a simple
question to answer." (2008:149)
         Before the term mathematics was conceived,"mathematics originated with the practical
problems of counting and recording numbers." From the ancient period, there were no verbal or
written words for numbers. They were intangible ideas relevant to few, enough, many, or too
much. Hence, counting was vague until it became concrete from simply knowing one and two to
counting with fingers and other objects for tracking. However, in the course of exchanging
commodities, the tasks became tedious at a point when goods were innumerable.
Representations were needed, thus, tallying was the response to this necessity, which evolved
from scratches on posts, knot-tying, to stick drawings. (Burton. 2007:1-2)
         Fast forward, mathematics literally meant "any subject of instruction or study" from the
Greek word mathemata. (Burton. 2007:1) But this denotation does not apply in today's sense
anymore. Students in all levels define it according to topics from arithmetic, algebra, geometry, to
higher level branches such as calculus. The assigned reading chapter quoted that it is the
"science of space and number…the science of patterns." (Grandgenett. 2008:150) With Koehler
and Misha's wicked problem, these patterns have evolved to a juncture where mathematics
should be realistic and feasible with the exponentially growing digitization.

Next segments: The Mathematics TPCK Teacher and What Makes TPCK Work in Mathematics

Posted by Irma Crespo on Tuesday, September 29th, 2009 12:50 PM

Hi, Midori:

You always bring up very interesting points. You made me realize that another view of
incorporating technology with pedagogy and content is engagement. Since children already have
preconceived ideas that mathematics is difficult upon entering the classroom, why not use
technology to stimulate their interest?

In EDD 569 (Reading Course), I thought, it's useful to make a lesson plan that will show the
concept of Algebra in another perspective after teaching them variables and constants so they
can find the relevance of Algebra in their own lives. I chose to integrate reading and music with
mathematics while using technology to show them that there is Algebra in the rhythm of music.
Whether or not they play instruments, at least, they will appreciate the fact that variables are not
only x's and y's, they can be represented in many different ways. I thought, it is TPCK combined
with Integration in other subject areas (reading, music, and kinesthetic), which caters to multiple

If anybody's interested, I have my complete lesson plan with addendums for example that also
includes the PowerPoint presentation and labyrinth game I created for interactive participation.

Irma Crespo                                         EDD 514                        October 2, 2009

Posted by Irma Crespo on Tuesday, September 29th, 2009 1:13 PM

Spreadsheets are indispensable productivity tools in the workplace. In our reading, Grandgenett
gave an example of a veteran teacher who had her students utilize this application. From this
model, I became aware that TPCK can also be a venue to prepare students for the workplace. If I
were to follow her approach, I can use spreadsheets in the unit on Formulas and Representations.
Given varied shapes and their corresponding formulas: triangles, squares, rectangles, rhombus,
cylinders, and circles, among others, they can create rows and columns for the measurements
and apply the formulas on the "formula bar," so they can get either the parameters or areas of the
shapes. Other instances are word problems where they can key in given information and create
their own formulas to generate the answers on the spreadsheet. With probability and statistics,
they can create their own graphs or charts to make predictions or determine a pattern.

Posted by Irma Crespo on Tuesday, September 29th, 2009 1:25 PM

Hi, Cynthia:

In the textile business, fractals are already being used to create prints for clothing designs. There
are fashion design softwares now that use fractals. What's amazing about these softwares is that
the user can choose just about any object and the program will miniaturize it in so many ways
repetitively since fractals are fractional geometry. These can be as simple as lines to the
elaborate patterns of the peacock feathers. Just an interesting fact to share:)

Posted by Irma Crespo on Wednesday, September 30th, 2009 11:54 AM

Hi, Midori:

I just sent you, Jennifer, and the Math group a zip file of the Algebra Labyrinth Lesson Plan with



Posted by Irma Crespo on Wednesday, September 30th, 2009 12:14 PM

Hi, Cynthia:

There's more about fractals from PBS. I actually watched one of Nova's episodes entitled Hunting
the Hidden Dimension. It's available online for viewing:


I hope this will pique your interest further.

Irma Crespo                                         EDD 514                        October 2, 2009

Posted by Irma Crespo on Wednesday, September 30th, 2009 1:07 PM

Why TPACK Works for Mathematics

-      It can combine with other methods or approaches in teaching and
     learning (adaptability is TPACK’s strongest point).
-   It can adjust to multiple learning abilities because its perspective of technology refers
     to both analog and digital besides being aware of ill structures and wicked problems.
-   The implementation is not tedious given the proper training and professional
-   The duration at which the success of teaching and learning becomes apparent is
    reasonable. A formative assessment during classes will give the teacher an inkling of
    how effective his/her TPACK approach is.
-    It is flexible in preparing students for standardized testing. The curriculum can be
    modified to fit state or national requirements.
-   Communicating the mathematics concepts, verbally and in writing, can be adjusted to
     specific learning needs. For example, technology can provide the sights and sound for
      the visual and hearing impaired.

These are all for now. Do you have other strengths to add on this list? Can you think of any
limitations or constraints with TPACK? If so, how do you fix these?

Posted by Irma Crespo on Wednesday, September 30th, 2009 2:41 PM

Check on this article, which sums up the ideas on integrating technology with content and
pedagogy. I believe, this is what TPACK is all about.

Promoting Appropriate Uses of Technology in Mathematics Teacher Preparation by Joe Garofalo ,
Hollylynne Stohl Drier , Suzanne Harper , and Maria A. Timmerman (University of Virginia), Tod
Shockey (University of Wisconsin—Stevens Point)


Dr. Duran, I added the pictures on this document. Thank you.

Posted by Irma Crespo on Wednesday, September 30th, 2009 4:39 PM

For those who do not have time to read the article I recommended: Promoting Appropriate Uses
of Technology in Mathematics Teacher Preparation, the following is a summary. I tried to explain
each example as simple as possible for those who are not in mathematics.

Guidlines on the appropriate use of technology in mathematics:

    - introduce technology in context
    - address worthwhile mathematics with appropriate pedagogy
    - take advantage of technology
    - connect mathematics topics
    - incorporate multiple representations” (2)

Irma Crespo                                    EDD 514                              October 2, 2009

Technology in Context
Example: Simulating Freefall With Parametric Quadratic Equations

“Derive an expression for the height of an object dropped from 500m above the surface of the
Earth, as a function of time.” Tool: Calculator
Quadratic expression: y = 500 – 4.9x

Student interpretation: outward downward, instead of straight down.
Equation set to Parametric: 500 – 4.9T

Students clearly recognize the freefall with parametric – straight down.

Benefit: With proper use of the calculator, students are able to visualize what the problem asks
them to simulate or re-create.

Worthwhile Mathematics with Appropriate Pedagogy
Example: Exploring the Pythagorean Theorem
                                                  2    2    2
Requires students to realize that a, b, and c of a + b = c are side lengths.

With the use of the software, Sketchpad, where a right triangle is created, regardless of side
lengths, which the students can manipulate, (a x a) + (b x b) = (c x c) = c . The visual aid in this
case is the squares created from each side of the triangle that clearly show the lengths of the
sides, which are also labeled with their calculated measurements.

Benefit: The students are encouraged to manipulate the side lengths of a right triangle and
discover themselves that the Pythagorean Theorem is valid. There is exploration through
technology for informal proof.

Taking Advantage of Technology
Example: Exploring Sierpinski Polygons

Irma Crespo                                   EDD 514                                October 2, 2009

Introduce fractals or fractional geometry to students. The students maneuver/play around with the
parameters of the already existing and well-known Sierpinski Polygons, specifically the ratios.
Starting from a triangle to the seemingly complex Sierpinski Polygons, which was simply created
by repetitively creating triangles using the interior midpoints of each triangle created.

The Sierpinski Polygon

With the Manipulated Ratio of the Sierpinski Polygon Set at 1.5

Benefit: Technology afforded the students the opportunity to learn functions, ratios, and similar
triangles. They are able to imagine and to visualize what happens when shapes are reduced to
miniature sizes repetitively: another concept to consider - iteration.

Connect Mathematics Topics
Example: Connecting Infinite Series and Geometric Constructions

Making students understand Infinite Series geometrically, which are supposedly two different
topics in mathematics.

The task necessitated students to visualize the series:

With Sketchpad, an equilateral triangle is first divided into four equal triangles using the internal
midpoints while rotating the color scheme clockwise as the students create another set of four
equal triangles within a triangle.

Irma Crespo                                 EDD 514                              October 2, 2009

Benefit: The students become aware of the fractional segments of the series visually and are able
to generalize that the series is equal to /3. Certainly, ¼ gets lesser and lesser in the series
because the triangles get smaller and smaller. Geometry helped in the understanding of Infinite

The other example in this aspect was connecting mathematics with Social Studies wherein the
students learn about margin of errors from Presidential Election results that were gathered from
the newspapers. A calculator was used to understand this.

Incorporate Multiple Representations
Example: Exploring Maximum Area

“Explore multiple representations involves a popular problem in which students find the maximum
area for a rectangular pigpen given a fixed amount of fencing.”

Using Microsoft Excel (a spreadsheet), the students key in the parameter given by the problem
and play around with the length of a side to find out the maximum.

Benefit: The students are able to represent the problem and solve it in multitude ways with
technology. They can tweak the parameter or the length of x while still satisfying the requirements
of the problem.

While there are more examples in the reading, I just took one application for each guideline to
exhibit how technology should be used based on these guidelines.

Irma Crespo                                           EDD 514                     October 2, 2009

 Also, I have pictures on my Word document. I tried to copy and paste them so you can see what
I'm talking about but it's not working. However, you can check the article and look at them since I
shortened and simplified the examples already.
Posted by Irma Crespo on Thursday, October 1st, 2009 5:46 PM

Hi, Midori:

I checked on the sites you provided. Unfortunately, some of the simulators required downloads.
My computer is in need of repair. At this time, I am using my sister's laptop. I can't take up so
much memory. I don't want her computer to slow down, either.

I like Learning by Seeing: Fun Visualization Tools That Educate.It shows what happens when
manipulating the different parts of a given formula (ex. What is needed to increase lift or to
maximize velocity?)

The EggMath is appealing for geometry especially for ellipses while the Famous Curves Applets
(this is where I wasn't able to run the applets) is an excellent way to manipulate variables for
visualization. I think, this latter application is catered more to students in AP classes or in math
undergrad. Most of them were covered, at least, at Calculus 2 level and higher Geometry such as

Posted by Irma Crespo on Thursday, October 1st, 2009 6:56 PM

Yeah, I was intimidated by the Applet Curves. I asked myself if I really have to go back and
review those! When I took them up, I literally "burned the midnight oil!" When I looked back at the
MI Content Expectations, the four years of math may include Geometry but not that high of a level.
At this stage, which is usually in the junior year, they are still learning how to make proofs.

I agree, as teachers, we have to be careful at choosing our sources. We both learned our lessons

Posted by Irma Crespo on Thursday, October 1st, 2009 6:18 PM

Following Dr. Duran's suggested format:

Technology in Context Example

Content: Quadratic Equations and Parametric Equations

Pedagogy: Comparing the Equations by Graphing

Technology: Calculator

Worthwhile Mathematics with Appropriate Pedagogy Example

Content: Pythagorean Theorem

Pedagogy: Making Informal Proofs

Technology: Sketchpad

Taking Advantage of Technology Example

Irma Crespo                                           EDD 514                      October 2, 2009

Content: Fractals

Pedagogy: Re-creating and manipulating the parameters of the Sierpinski Polygon

Technology: MicroWorlds

Connect Mathematics Topics Example

Content: Infinite Series

Pedagogy: Geometry Integration - Visualizing Number Series with Geometry

Technology: Sketchpad

Incorporate Multiple Representations Example

Content: Maximum Areas

Pedagogy: Representations

Technology: Spreadsheet
Posted by Irma Crespo on Thursday, October 1st, 2009 6:47 PM

Hi, Erica:

Integrating technology in mathematics is not only for demonstration and reinforcement. There are
numerous reasons to incorporate technology. One of these is keeping the students at par with the
changing generation. Certainly, we wouldn't want them using a typewriter when state-of-the-art
word processors are widely utilized and are readily available. Should we keep them away from
learning if they are unable to attend school when there's video-conferencing at hand? They can
be physically absent but virtually present with other students in class using this technology. As
math teachers, do we painstakingly ask the students to solve large numbers or graph infinite
series by hand when there are calculators or softwares such as Sketchpad? These all boil down
to the basic role of technology - make tasks more effective and more efficient. In teaching, it's to
make learning more meaningful.

Posted by Irma Crespo on Thursday, October 1st, 2009 7:26 PM

Hi, Marguerite:

Just as Midori and Jennifer indicated there no content standards for integrating other subject
areas but there are studies out there on "subject area integration." The plus side about it is
preparing the students to think globally. The downside of it is that teachers specialize in certain
subject areas and incorporating other curriculum subjects means understanding them, too. For
instance, a math teacher who is introducing the concept of functions and relations with the
science of mass and gravity also has to get acquainted with the concept of the latter in order to
make connections. It's another learning hurdle the teacher has to make.

I made a lesson plan on incorporating literature and technology with mathemtics. This is using the
Harry Potter series in developing problem solving skills since the books offer a lot of opportunities

Irma Crespo                                           EDD 514                     October 2, 2009

to think and do mathematics. There's the concept of currency exchange with Harry Potter's
galleons, knuts, and sicles or the ideas on proportions with Harry Potter potions, or the geometry
of angles in analyzing Harry Potter's Quidditch games, and many more. With these, the students
are encouraged to use varied forms of technology to demonstrate the concept of the word
problems and their corresponding solutions. Besides the multitude mathematical scenarios found
in Harry Potter books, the students can readily connect since it's a best seller and kids love to
read them. In my rationale, students think the math, represent the math, communicate the math,
apply the math in new situations, and choose the proper technology for the math. My grading tool
here is a rubric.

Posted by Irma Crespo on Thursday, October 1st, 2009 7:33 PM

Correction: "Just as Midori and Jennifer indicated there are no content standards..."

TPACK bridged the gap between the literature, the mathematics, and the learning.

Posted by Irma Crespo on Thursday, October 1st, 2009 11:22 PM

Hi, Midori, Jennifer, and the Math Group:

If you're wondering why the fractions are not simplified on the Algebra of Music, it's because
simplifying them changes the measures, thus, the tempo. Musicians/composers need to know
beats per measure - this determines how fast or slow the music is. Thus, the mood it conveys.
This is where students apply Algebra in the proper context. Besides, the main purpose of the
lesson is not really the fraction but the representation of variables in Algebra. This is where
Koehler and Misha's wicked problem comes in - when content is taught in varied contexts, which
provide a more open mind and understanding of applications.

Posted by Irma Crespo on Friday, October 2nd, 2009 11:09 AM

Also, since the lesson plan is geared for high school students, the Algebra in context can be a
tool for critical thinking. What happens to the beats per measure when the fractions are simplified?
Why do you think this change occurs? These are older kids and they'll probably find out even
before these questions are raised.

Posted by Irma Crespo on Friday, October 2nd, 2009 9:06 AM

Marguerite. Thanks. It's sad, though. I have so many lesson plans in my portfolio but have not
implemented them yet. So, I haven't seen those faces light up (or the opposite) yet.

We read all the Harry Potter books. Infact, there's so much potential in literature that there are
multitude of opportunities for teachers to create their own word problems. I have also read all the
books in the following series: Twilight, Inkheart, Narnia, and Rodin's. There are more to read!

Posted by Irma Crespo on Friday, October 2nd, 2009 10:12 AM

Here's the link for High School Math Standards:


       Irma Crespo                                  EDD 514                               October 2, 2009

                                  Participation in Online Discussions Rubric

   CATEGORY          4                        3                        2                        1
Active               Participates actively    Participates actively    Sometimes                Seldom participates
participation        in discussion.           in discussion.           participates actively    actively in
                     Responds to others'      Responds to others'      in discussion.           discussion.
                     ideas and questions.     ideas and questions.     Responds to others'      Responds to others'
                     Poses thoughtful and     Sometimes poses          ideas and questions.     ideas and questions
                     relevant new             questions. Enters        Enters discussion        in a quick and
                     questions. Enters        discussion mid-week      late-(Thursday).         superficial manner.
                     discussion early.        (Wednesday)                                       Enters discussion
                                                                                                late (Friday).
Makes Connections Makes relevant and          Sometimes makes          Rarely makes             Does not make
                  insightful                  connections between      connections between      connections between
                  connections between         the text and own         the text and own         the text and own
                  the text and own            current or previous      current or previous      current or previous
                  current or previous         experience, other        experience, other        experience, other
                  experience, other           texts, current events.   texts, current events.   texts, current events.
                  texts, current events.
Asks Questions       Poses thoughtful         Poses questions.         Poses questions.         Rarely poses
                     questions. Grapples      Tries to answer          Answers to               thoughtful questions.
                     with these questions.    these questions.         questions are            Answers to
                     Offers an                Sometimes offers an      superficial.             questions are
                     interpretation or        interpretation or                                 superficial.
                     analysis of the          analysis of the
                     selection.               selection.
Uses Examples        Returns to the text      Returns to the text      Rarely returns to the    Does not return to
from Text            often to cite specific   sometimes to cite        text sometimes to        the text to cite
                     examples. Provides       specific examples.       cite specific            specific examples.
                     specific evidence        Demonstrates             examples.                Does not
                     that demonstrates        familiarity and          Demonstrates some        demonstrate
                     familiarity and          understanding of the     familiarity and          familiarity and
                     understanding of the     assigned reading         understanding of the     understanding of the
                     assigned reading         selection.               assigned reading         assigned reading
                     selection.                                        selection.               selection.
Reflects on Practice It is evident that the It is sometimes       There is weak         There is no evidence
                      student is reflecting evident that the      evidence that the     that the student is
                      thoughtfully on their student is reflecting student is reflecting reflecting
                      beliefs and teaching thoughtfully on their thoughtfully on their thoughtfully on their
                      practice. Growth is beliefs and teaching beliefs and teaching beliefs and teaching
                      evident.              practice.             practice.             practice.
       Adapted from


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