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Master of Arts in Teaching Mathematics
Student Portfolio
for
Agamyrat Altiyev
Spring 2014
MATM Program: End-of-Semester Reflection
Name of Student: Agamyrat Altiyev
Semester ending / just ended: SPRING 2014
Thank you for taking the time to write this important reflection on your semester – it will be very
valuable to our ongoing efforts to assess and improve the MAT-M program. It will be especially
helpful to us if most of your answers can include a reference to one or more documents inserted
in your portfolio. Please be as thoughtful and honest as you can in your reflections – only
people that are part of the program improvement and evaluation will see them.
1. What do you think was the most important thing you did/learned in the program this semester? Why?
I learned several great things from MAT-M program this semester as of my first semester.
The most important thing was that program gave me a chance to apply the knowledge that
I learned to my teaching which is so hard to find and achieve it. MAT-M courses were live
reference/research tool that I learn from highly qualified professors. It was a like semester
long professional development but better than them because I learned and applied during
the courses.
2. Do you feel you have improved your ability to use technology and to communicate effectively, both
orally and in writing? If yes, how? If no, why do you think that is and what changes might be
helpful?
I was an IT Coordinator before I started teaching, so I felt confident on using technology
in my classroom. Technology in Math Classroom course taught me more useful types of
technology to use and improved my ability and skills to use it on daily bases. Several
software programs such as GeoGebra teaches and entertains students more than a teacher
does sometimes. Now, I use GeoGebra more often in my classroom. My math
communication skill developed on every day of my class visit to UTEP, because I heard
and learned proper content language that many school teachers doesn’t learn unless they
research.
3. What have you learned this semester in the MAT-M program that connects to the math you are or will
be teaching?
As I expressed above, GeoGebra was best tool that I learned and improved this semester. I
used the software both for Geometry and Algebra. GeoGebra is such an educationally rich
tool that students can use it and observe different aspects of a subject.
4. What key mathematical knowledge/ideas have you learned?
In my Algebra and Number Theory Course, I learned and understood logically how prime
numbers are can be infinite resulted better understanding of Fundamental Theorem of
Arithmetic. Although, I learned this before, now I have broader knowledge connection
between concepts.
5. Have your mathematical habits of mind (as outlined at http://www.corestandards.org/math/practice)
improved? If yes, how? If no, why do you think that is and what changes might be helpful?
My understanding of mathematical habits of mind is that math is a tool for a person to use
when its required but the person should has an ability to use it by previously learned
knowledge. It shouldn’t require a memorization of formulas or mnemonic which results to
be forgotten after a period of time. If this is definition of mathematical habits of mind, I
definitely improved and introduced such kind of teaching in my classroom.
MATM Program: End-of-Semester Reflection
6. Has your capacity to conduct mathematics education research (in your classroom or beyond)
improved? If yes, how? If no, why do you think that is and what changes might be helpful?
I’ve been researching since I started teaching on every aspect of teaching including
classroom management, content based research or articles of secondary school math. I
learned applied research during this semester where I learn and apply the research which is
better than reading a research paper. Now, while I’m conducting mathematics education
research, I use software tools to apply it in different ways and give it varied taste than it
was written on a research.
7. Have your perspectives of mathematics teaching and learning changed this semester?
If yes, how? If no, why do you think that is and what changes might be helpful?
I joined MATM after 6 month officially teaching a math classes at both middle and high
school. I didn’t judge some nonsense teaching tools or topics that we are forced to teach in
math classes. Now, I’ve some confidence on what is important to teach students and what
is not. I learned how to make better comprehensive connection between different level
topics in math classes. I learned not to rely on a specific teaching material but use varied
resources and choose best one to address students.
8. Do you feel more prepared to teach students to think mathematically and understand conceptually?
If yes, how? If no, why do you think that is and what changes might be helpful?
I do feel more prepared to challenge students to think mathematically and let them
understand conceptually by themselves, but I need more practice on applying such a
teaching style. Math knowledge and software tools that I learned made me confident to use
different representations to let students think mathematically and deeper exercise on such
tools would be more helpful such as I will be learning in Applied Mathematics course in
Fall 2014 semester.
9. What other comments do you have?
Thank you for making me to rehearse most of the things happened during Spring 2014
semester and how actually it succeed my academic life.
MATH 5365
Software Review: PSTricks
Agamyrat Altiyev
March 6, 2014
1 Exercises
1.1 Archimedes’ demonstration of formula for the area of a circle
Make a diagram like the one that follows, illustrating Archimedes’s demonstration of the formula forthe area of a circle.
r
∼ πr
A = (πr)r = πr2
r
1.2 Plot the function
0
1
2
3
4
0 1 2 3 40
−1
−2
−3
−4
0−1−2−3−4
y = x2
y = −x2
1
1.3 Petersen Graph
Make a diagram of the Petersen graph, with ten vertices and fifteen edges, as below. Color the verticeswith three colors so that no connected vertices are the same color.
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
2
\documentclass{article}\title{Software Review: PSTricks}\author{Agamyrat Altiyev}\date{March 6, 2014}\usepackage{pstricks}\usepackage{pst-node}\usepackage{pst-plot}\setlength{\topmargin}{-1.74cm}\setlength{\headheight}{1cm}\setlength{\headsep}{0.74cm}\setlength{\textheight}{24.62cm}\setlength{\textwidth}{15.5cm}\setlength{\oddsidemargin}{0.46cm}\setlength{\evensidemargin}{0.46cm}
\begin{document}\maketitle
\section{Exercises}
\subsection{Archimedes' demonstration of formula for the area of a circle}Make a diagram like the one that follows, illustrating Archimedes's demonstrationof the formula for the area of a circle.\\\\\begin{pspicture}(-1,-1)(18,5)\\\pscircle(2,2){2.25}\psline(-0.25,2)(2,2)(4.25,2)\psline(2,-.25)(2,4.25)\psline(.9,0.05)(2,2)(3.1,3.95)\psline(0.1,.85)(2,2)(3.9,3.15)\psline(0.05,3.05)(2,2)(3.95,.9)\psline(.85,3.9)(2,2)(3.1,0.05)\pscurve (5,4)(5.5,4.05)(6,4)(6.5,4.05)(7,4)(7.5,4.05)(8,4)(8.5,4.05)(9,4)(9.5,4.05)(10,4)(10.5,4.05)(11,4)\psline (4.5,2)(5,4)(5,4)(5.5,2)(6,4)(6.5,2)(7,4)(7.5,2)(8,4)(8.5,2)(9,4)(9.5,2)(10,4)(10.5,2)(11,4)\pscurve (4.5,2)(5,1.95)(5.5,2)(6,1.95)(6.5,2)(7,1.95)(7.5,2)(8,1.95)(8.5,2)(9,1.95)(9.5,2)(10,1.95)(10.5,2)\psline (11.5,2)(11.5,4)\put(12,3){$r$}\psline (4.5,1.5)(10.5,1.5)\put (7.5,1) {$\sim\pi r$}\put (6.5,0) {$A=(\pi r) r = \pi r^2$} \put (3.2,1.7) {$r$}\end{pspicture}\\
\subsection{Plot the function}\vspace{.5 cm}\begin{pspicture}(-4,-4)(4,4)\psaxes{->}(0,0)(4.25,4.25)\psaxes{->}(0,0)(-4.25,-4.25)\psplot[plotpoints=2500]{0}{2}{x 2 exp}\put (2.25,3){$y=x^2$}\psplot[plotpoints=2500]{-2}{0}{ x 2 exp neg}\put(-3,-2.5){$y=-x^2$}
1
\end{pspicture}
\subsection{Petersen Graph}Make a diagram of the Petersen graph, with ten vertices and fifteen edges, asbelow. Color the vertices with three colors so that no connected vertices are thesame color.\vspace{1 cm}
\begin{pspicture}[showgrid](-5,-5)(5,5)\psset{radius=0.25}
%inner shape:star\put(1.12,-.22){\Circlenode[fillstyle=solid,fillcolor=red]{1}}\put(0.12,-1.22){\Circlenode[fillstyle=solid,fillcolor=red]{2}}\put(-1.14,-.58){\Circlenode[fillstyle=solid,fillcolor=black]{3}}\put(-.92,.82){\Circlenode[fillstyle=solid,fillcolor=black]{4}}\put(0.48,1.04){\Circlenode[fillstyle=solid,fillcolor=yellow]{5}}\ncline{1}{4}\ncline{1}{3}\ncline{2}{4}\ncline{2}{5}\ncline{3}{1}\ncline{3}{5}
%outer shape: polygon\put(3.5,-.62){\Circlenode[fillstyle=solid,fillcolor=yellow]{6}}\put(.49,-3.61){\Circlenode[fillstyle=solid,fillcolor=black]{7}}\put(-3.28,-1.68){\Circlenode[fillstyle=solid,fillcolor=red]{8}}\put(-2.61,2.51){\Circlenode[fillstyle=solid,fillcolor=yellow]{9}}\put(1.58,3.16){\Circlenode[fillstyle=solid,fillcolor=red]{10}}
\ncline{1}{6} \ncline{6}{7}\ncline{7}{8}\ncline{9}{8}\ncline{9}{10}\ncline{6}{10}\ncline{2}{7}\ncline{3}{8}\ncline{4}{9}\ncline{5}{10}
\end{pspicture}
\end{document}
2
Angles, Arc, Secant and Tangent
Agamyrat Altiyev
May 1, 2014
Agamyrat Altiyev ([email protected]) Angles, Arc, Secant and Tangent May 1, 2014 1 / 7
Lines that Intersect Circles
A chord is a segment whose endpoints lie on a circle
A secant is a line that intersects a circle at two points.
A tangent is a line in the same plane as a circle that intersects it atexactly one point.
Agamyrat Altiyev ([email protected]) Angles, Arc, Secant and Tangent May 1, 2014 2 / 7
Constructing on GeoGebra
Exploring Radii ⊥ to chords
Use GeoGebra to construct a circle A.
Use segment tool to draw a radius AB in circle.
Use Perpendicular Line tool to draw a chord (that is not a diameter)that is perpendicular to radius. Label C and D as endpoints ofchords. Label point of intersection, point E .
Use distance tool to find the segment length of EC and DEseperately.
What appears to be similar?
Hold and drag intersection point E on the radius.
Summarize your observation next to the figure by using Text tool.
Agamyrat Altiyev ([email protected]) Angles, Arc, Secant and Tangent May 1, 2014 3 / 7
Definition
Exploring Tangent Lines A tangent line of a circle is perpendicular to itsradius.
Construct a circle by using circle tools.
Mark two points on circle with labels B and C .(Points should not beplaced as endpoints of diameter)
Draw radii AB and AC .
Draw a line perpendicular to radius AB and AC .
Put a point on intersection of two lines labeled D.
Find the distance of segment BD and CD.
Hold and drag intersection point to any directions and observe thechange of distance.
Summarize your observation next to the figure by using Text tool.
Agamyrat Altiyev ([email protected]) Angles, Arc, Secant and Tangent May 1, 2014 5 / 7
Exploring Tangent Lines
Pacman
Playing with tangent lines makes PACMAN eat your point.
Agamyrat Altiyev ([email protected]) Angles, Arc, Secant and Tangent May 1, 2014 6 / 7
End of Presentation
THANK YOU FOR WATCHING!
Agamyrat Altiyev ([email protected]) Angles, Arc, Secant and Tangent May 1, 2014 7 / 7
MATH 5370
Fermat’s Last Theorem
Pierre de Fermat (1601-1665)
• Born Beaumont de Lomagne
• Fermat Secondary School
• “Fermat enfant de la Lomagne”
• Statue of Fermat in town center
Fermat’s Life
• Father was second consul of the town
• Mother brought social status of the parliamentary noblesse de robe.
• By 1631, lawyer and bought offices of conseiller in parlement in
Toulouse
• Change his name from Pierre Fermat to Pierre de Fermat
• Married his mother’s cousin Louise de Long and had 5 kids
Fermat’s letters
• Fermat did not put his work in polished form
for publication
• Majority of Fermat’s work was published
posthumously
• Instead he would send letters with other mathematicians about his discoveries without proof
• In a manner of challenging others to figure out the results he had already obtained
• Many people became annoyed by his claims partly because most of his claims were nearly impossible to prove
• Many even felt like Fermat was just teasing them
•
Mathematical History of the theorem
Mathematical History of the theorem
1. Babylonians (1800-1650 B.C)
- Plimpton 322
(1:)59:00:15 1:59 2:49 1
(1:)56:56:58:14:50:06:15 56:07 1:20:25 2
(1:)55:07:41:15:33:45 1:16:41 1:50:49 3
(1:)53:10:29:32:52:16 3:31:49 5:09:01 4
(1:)48:54:01:40 1:05 1:37 5
(1:)47:06:41:40 5:19 8:01 6
(1:)43:11:56:28:26:40 38:11 59:01 7
(1:)41:33:45:14:03:45 13:19 20:49 8
(1:)38:33:36:36 8:01 12:49 9
(1:)35:10:02:28:27:24:26 1:22:41 2:16:01 10
(1:)33:45 45 1:15 11
(1:)29:21:54:02:15 27:59 48:49 12
(1:)27:00:03:45 2:41 4:49 13
(1:)25:48:51:35:06:40 29:31 53:49 14
(1:)23:13:46:40 56 1:46 15
Continued (history)
•
Continued (history) •
Continued (history)
•
Continued (history)
•
Why not cubes?
Perfect Cubes:
13 = 1
23 = 8
33 = 27
43 = 64
53 = 125
63 = 216
73 = 343
83 = 512
93 = 729
103 = 1000
113 = 1331
• Elliptic torch
Different Representations of the Theorem
Modular Form
Video animation-1990
Wolfram Demonstration
• Every elliptic curve is a modular form in disguise.
• Taniyama-Shimura conjecture(now theorem) was a bridge between modular forms with many symmetries and elliptical curve (donuts)
• Two different worlds lives on two different universes and this theorem is a bridge to it.
• Many mathematicians claims that this conjecture is impossible to prove.
• Did Taniyama/Shimura’s conjecture had anything to do with Fermats theorem?
• Frey’s idea : (by contradiction)
• If Fermat is wrong so does Taniyama, if Tanimaya is right so does fermats last theorem
Video animation-Modular Theorem
•
Andrew Wiles(b. 04/11/1953)
Wiles personal struggles with FLT •
MTED 5326
Agamyrat Altiyev
MTED 5326
Dr. Carrejo
5/7/14
Within my teaching philosophy, I agree to the statement that all math students are language
learners and I also think that it depends on teaching style and philosophy as well. Some teachers are
using content loaded style where students have to learn all this new terms, formulas, symbols and more,
but some teachers might use more spoken language style where students doesn’t have to learn a lots of
new terms or symbols but understand the language.
As an ELL student, I never struggled on math as native language speakers does. Think of a 12
years old student who never spoke or learnt English, but first semester of 6th grade school year taking
20 hours per week English courses, and on second semester starts to take basic Math classes in English
where he never studied math in English. That was me and country was Turkmenistan. I didn’t struggle
on math during high school years because my math teachers taught me how to see the math picture
behind written words and don’t get stuck on meanings of certain words that doesn’t help you to figure
out the solution except confusion.
Even though, I agree that mathematics is a language but everybody is grown up with it. Nobody
is exposed to mathematics language after certain years of age. Even early humankind whom didn’t know
how to count used relations concept of mathematics to keep track of numbers of their animals. Now,
starts counting earlier ages and learn how to do operation before kindergarten, but they are never told
that: “You are learning a new language called math.”
Since math is part of everyday life since born to death, I don’t quietly agree with the term that
ALL math students are language learners. In this case, my concern is firstly limiting math to students and
secondly being a language learner which doesn’t precede high school. A math is such a language where
you learn a new word (term, technique, and etc) almost every day and you wonder how impressive that
new word is.
I also support that pedagogy that works for math students should also work for ELL’s as well.
This also rely on art of math teachers since they have to use some extra tools (gestures, drawings,
actions with objects) to support ELLs’ success in math. Teachers should pull upon these extra tools as
they launch and discuss tasks, and students should also be encouraged to use these tools to represent
themselves. Teachers should also use pedagogical practices of organizing lessons and scaffolding
academic language to support ELL’s.
Historical math that we learnt in this class is one of a fundamental tool to introduce, explain and
practice a new topic in math classroom for every kind of students including ELL’s. We all needed was
math tools not a language to understand the given problem. ELL’s should be cleared that their English
inefficiency is not a boundary to their success in math or appreciation to math.