MATHEMATICS AND INSTRUCTIONAL TECHNOLOGY
Department of Urban Education
21.300.348, 3 credits
Thursdays, 11:30 to 2:20
Education Lab, 148 Bradley Hall; Computer Lab, 409
Bradley Hall
Instructor: Dr. Arthur B. Powell abpowell@andromeda.rutgers.edu
Office: 156 Bradley Hall;
Phone: 973.353.3530
Office hours: Tuesday 1:00
– 2:00 pm, Thursdays 10:00 – 11:00
Teaching Assistant: F. Frank
Lai fflai@eden.rutgers.edu
Office: 178 Bradley Hall;
Phone: 973.353.3538
Over
the millennia, humanity has witnessed radical, technological and pedagogical
changes in mathematics education.
These include corresponding changes in technologies for representing as
well as manipulating ideas and in pedagogies for teaching mathematics as well
as participating social groups in mathematics. From about 3,000 BC, with the invention of writing in the
form of Egyptian hieroglyphs and Near Eastern cuneiform, mathematics teaching
has essentially involved the following:

variations of choral
response;

memorization of rules,
facts, and procedures; as well as

subjectcentered
curricula in a teachercentered environment.
However,
recent, important developments have dramatically changed the way that educators
think about the teaching of school mathematics. Since the beginning of the 1950s, the mathematics education
community has progressively focused more pedagogical attention on ways of
making school mathematics meaningful as well as encouraging the serious
participation in mathematics of increasingly larger and diverse proportions of
the countryˆïs students. This focus
has prompted pedagogical shifts away from concentrating on algorithms and
computations toward emphasizing

studentcentered
curricula with teacher as facilitator or coach;

discovery activities,
openended investigations; and

sense making.
Paralleling
this pedagogical shift, electronic, information technologies have advanced
rapidly as have their use in research mathematics and in mathematics
education. This technological
advance is widespread and has reached a point where sophisticated educational
hardware and software tools are available to mathematics classes even in
economically poor, urban schools.
These technological tools have simultaneously made obsolete many of the
algorithms taught in previous mathematics classes (e.g., calculating square
roots by hand and interpolating logarithms) and made possible many previously
impractical activities (such as simulations of large samples of empirical
probabilities to acquire insight into theoretical probabilities and graphing
exponential equations to explore effects of varying parameters).
These
twin changes have contributed to the now prominent roles played by
meaningfulness and technology in many school mathematics curricula and
classrooms. More to the point,
mathematics educators are encouraged (even urged) to use technological devices
as tools to enhance studentsˆï ability to develop meaningfully their
mathematical ideas and ways of reasoning mathematically. For instance, in 2000, the National
Council of Teachers of Mathematics (NCTM) published what has become an
influential document, Principles
and Standards for School Mathematics
(http://standards.nctm.org/document/index.htm).
In
this volume, the NCTM describes principles that all mathematics classrooms
should meet and specifically lists the use of technology as one of their six
principles and further declares that all school mathematics courses should regularly and meaningfully employ
technology. The challenge for
prospective and practicing teachers alike is to determine how to incorporate
technology sensibly and effectively into the learning and teaching of
mathematics.
The
central goal of this course is to enable you to learn how technology can be
used to enhance mathematics learning and acquire facility in using particular
technologies as an effective pedagogical tool in the mathematics courses that
you will teach. This course is
organized around the exploration of concepts in three strands of mathematics:
number and algebra; geometry; and data analysis and probability. In these strands, using various technological
tools, you will revisit your own learning of mathematics and investigate
mathematical concepts through collaboration, problem solving, and mathematical
justification.
More
specifically, by participating actively and completing assignments and projects,
you will accomplish the following objectives:

Revisit, broaden, and
deepen your understanding of certain mathematical ideas and forms of
mathematical reasoning through the use of technology.

Extend your facility to
use various types of electronic technologies available to enhance the teaching
of school mathematics.

Enlarge your knowledge
of classroom, problemsolving activities that incorporate technology.

Increase your
understanding of psychological and educational principles that underpin
effective uses of technological tools in mathematics classrooms.

Augment your awareness
of the racial, gender, and economicclass imperatives for access to
technologyrich, learning environments for learning mathematics.

Develop your ability to
design lessons that incorporate technology as well as your facility to evaluate
critically lessons that involve technology.
To achieve the goal and objectives of the course, you
can expect to work outside of class in a neighborhood of eight to ten hours per
week on assignments and projects.
In class, you will continually engage in critical discussions and
reflections about your mathematical learning and about the learning and
teaching of mathematics with technology as well as about issues of equity and
access to technologyrich mathematics classrooms.
This
course is part of the sequence of courses required for elementarymiddle school
New Jersey State certification.
21.300.342
Elementary Mathematics and Pedagogy
21.300.343
Elementary Science
By
permission
By
the end of this course, you should be able to perform the following skills with
the listed technology:
Microsoft
Word
TI73
Explorer Calculator
1.
Use Constant function
with countercreating number patterns
2.
Perform basic
computations (including permutations, combinations, factorial)
3.
Use List editor to enter
categorical and numerical data
4.
Use Plot to specify a
scatter plot and a histogram
5.
Use Graph and Window to
display List data plotted in an appropriate viewing window
6.
Use Y=, Window, and
Graph to display functions using an appropriate viewing window
7.
Use the zoom and trace
features in the graphing application
8.
Create a split screen
9.
Create a table of values
10. Attach formulas to list names (formulas should include
the use of the sequence command)
11. Compute univariate and bivariate statistics
12. Find an appropriate regression equation for a set of
data
13. View a set of data and a regression equation
simultaneously
TinkerPlots
(for elementary education students)
Fathom
(for secondary education students)
1. Enter Data in either a New Collection or a New Case
Table with appropriate Attribute Names
2. Import Data from an Internet website into a either a
New Collection or a New Case Table
3. Be able to Lock and Unlock a Collection
4. Display Univariate Data in appropriate graphs (e.g.,
Dot plot, Line Graph, Histogram, Box Plot)
5. Display Bivariate Data in a Scatterplot, Add a
LeastSquares line, and Create a Residual Plot
6. Create a Summary Table with the Basic Statistics
calculated for a data set
7. Add a formula to the Summary Table to display another
Statistic (e.g., variance)
8.
Enter data in Collection or Case Table based on
a Formula that uses at least one built in Function
9. Create a Function Plot with a New Plot and Enter a
Formula for the Function you wish to graph.
10. Add a New Slider and use that Variable (Global Value)
in a Formula, especially a formula that controls a function plot.
The
Geometer's Sketchpad, version 4
1.
Create basic geometrical
objects: Point, line segment, line, ray, circle, polygons, ˆâ
2.
Construct geometrical
objects: Midpoint of a segment, a line passing through a point and
perpendicular to a segment, a line passing through a point and parallel to a
segment or line, connecting midpoints of segments, ˆâ
3.
Particular types of
triangles (acute, obtuse, right, scalene, isosceles)
4.
Particular types of
quadrilaterals (parallelogram, rectangle, rhombus, square, kite)
5.
Centers of triangles:
incenter, circumcenter, orthocenter, centroid
6.
Arcs of circles
7.
Perform operations:
Apply transformations to geometrical objects (translation, reflection,
rotation, and dilation)
8.
Measure lengths and angles
9.
Perform computations
using the calculator tool
10. Hide objects
11. Label objects
12. Create captions and apply formatting options to text
13. Include animation
14. Create custom tools (version 4)
15. Plot points and create graphs
Probability
Explorer
Microsoft
Excel
1. Mathematics Education Bundle, published by Key
Curriculum Press <http://www.keypress.com/>,
a speciallypriced, student package: for elementary and middle—The
Geometer's Sketchpad and TinkerPlots ($66.75) or for secondary—The
Geometer's Sketchpad and Fathom 2 ($66.75).
2. New Jersey Core Curriculum Content Standards for
Mathematics <http://www.state.nj.us/njded/cccs/s4_math.htm>
3. New Jersey Professional Standards for Teachers and
School Leaders <http://www.state.nj.us/njded/profdev/profstand/standards.pd>
4. Zinsser, W. (1988). Writing mathematics. In Writing
to learn (pp. 149167). New York:
Harper and Row.
5. Powell, A. B. (2001). Capturing, examining, and
responding to mathematical thinking through writing. Pythagoras (55), 38.
6. Huinker, D. (2002). Calculators as learning tools for
young children's exploration of number. Teaching Children Mathematics, 8(6), 316321.
7. Kieran, C., & Guzmán, J. (2005). Five steps to
zero: Students developing elementary number theory concepts when using
calculators. In W. J. Masalski & P. C. Elliott (Eds.), Technologysupported
mathematics learning environments
(Vol. SixtySeventh Yearbook). Reston, VA: National Council of Teachers of
Mathematics.
8. Powell, A. B. (1993). Pedagogy as ideology: Using
Gattegno to explore functions with graphing calculator and transactional
writing. In C. Julie, D. Angelis & Z. Davis (Eds.), Proceeding of the
Second International Conference on the Political Dimensions of Mathematics
Education (pp. 356369). Cape Town:
Maskew Miller Longman.
9. Zevenbergen, R. (2000). "Cracking the code" of
mathematics classrooms: School success as a function of linguistic, social, and
cultural background. In J. Boaler (Ed.), Multiple perspectives on
mathematics teaching and learning
(pp. 201223). Westport, CT: Albex.
10. Powell, A. B. (2004). Investigating Your Mathematical
Thinking: Calculator Explorations and Report Writing.
11. Asimov, I. (1982). Numbers large and small. Natural
History.
12. Peressini, D. D., & Knuth, E. J. (2005). The role
of technology in representing mathematical problem situations and concepts. In
W. J. Masalski & P. C. Elliott (Eds.), Technologysupported mathematics
learning environments (SixtySeventh Yearbook, pp. 277290). Reston, VA:
National Council of Teachers of Mathematics,
13. McGraw, R.,
& Grant, M. (2005). Investigating mathematics with technology: Lesson
structures that encourage a range of methods and solutions. In W. J. Masalski
& P. C. Elliott (Eds.), Technologysupported mathematics learning
environments (Vol. SixtySeventh Yearbook, pp. 303317). Reston, VA: National
Council of Teachers of Mathematics.
14. Bakker, A., & Frederickson, A. (2005). Comparing distribution
and growing samples by hand and with a computer tool. In W. J. Masalski &
P. C. Elliott (Eds.), Technologysupported mathematics learning environments (Vol. SixtySeventh Yearbook). Reston, VA: National
Council of Teachers of Mathematics.
15. Ploger, D., Klingler, L., & Rooney, M. (2000).
Spreadsheets, patterns, and algebraic thinking. In B. Moses (Ed.), Algebraic
thinking, grades K12: Readings from NCTM's schoolbased journals and other
publications (pp. 232237). Reston:
National Council of Teachers of Mathematics.
16. Clements, D. H. (2003). Teaching and learning
geometry. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A
research companion to Principles and Standards for School Mathematics (pp. 151178). Reston, VA: National Council of Teachers
of Mathematics.
17. Shaughnessy, J. M. (2003). Research on students'
understanding of probability. In J. Kilpatrick, W. G. Martin & D. Schifter
(Eds.), A research companion to Principles and Standards for School
Mathematics (pp. 216226). Reston,
VA: National Council of Teachers of Mathematics.
18. Jones, G. A., Thornton, C. A., Langrall, C. W., &
Tarr, J. E. (1999). Understanding students' probabilistic reasoning. In L. V.
Stiff (Ed.), Developing mathematical reasoning in grades K12, 1999 Yearbook
of the National Council of Teachers of Mathematics (pp. 146155). Reston: National Council of Teachers
of Mathematics.
19. Tang, E. P., & Ginsburg, H. P. (1999). Young
children's mathematical reasoning: A psychological view. In L. V. Stiff (Ed.), Developing
mathematical reasoning in grades K12, 1999 Yearbook of the National Council of
Teachers of Mathematics (pp. 4561).
Reston: National Council of Teachers of Mathematics.
20. National Library of Virtual Manipulatives for
Interactive Mathematics: http://matti.usu.edu/nlvm/nav/index.html
Readings,
problem solving, and written reflections 20%
Portfolio of
selected, completed assignments 30%
Project and
showcase presentation 30%
Final
Examination 20%
Date 
Mathematics Topics 
Technology Tools 
Readings and Assignments 
Week 1: Introduction 

Week 1 
From Movement to Algebra;
Writing as a Technological Tool for Learning Mathematics 
Paper
and pencil 

Task: Leapfrog (or
Chip Switch) Situation 
Write about your
experience with the Leapfrog (or Chip Switch) Situation and how you used
patterns to develop a general solution. 
Read Zinsser (1988)
and Powell (2001) as well as write an abstract and a reflection paper for
each article. 
Through the classˆïs
electronic list on Blackboard, distribute a brief assessment of your
experience writing in connection with learning mathematics. 
Weeks 25: Number and Algebra 

Week 2 
Generalizing and specializing;
Writing as Learning; counting activities with Unifix cubes. 
Microsoft Word: Table,
Equation Editor, and Graphic Tools 

Discussion Questions
for Reading (DQR) 
Generalizing and
specializing in the Leapfrog Problem 
Task: Find a number
with exactly n factors: A
discussion of Ian Childˆïs solution expressed in prose. 
Writing, reflecting,
and learning 
Find an article about
the use of writing in mathematics classes in elementary, middle, or secondary
school; write an abstract and a commentary; and distribute them to the class
through Blackboard. 
Using MS Word, create
a table containing data from the Leapfrog Situation, including algebraic
expressions for the case of n
frogs per side and submit through Blackboard. 
Read Huinker (2002),
write an abstract, a reflection paper, and answer Week 3 DQR. 
Week 3 
Combinatorics; Geometric
and numerical patterns of multiplication and division; inverse operations,
and factors; square, triangular, and prime numbers 
TI73 Explorer 

Discuss reading and
DQR 
Tasks: (1) Five Steps
to Zero and (2) Towers Problem (4tall, 2colors) 
Problemsolving
heuristics: tables and simplifying a problems 
Write a letter to a
student who is ill and unable to come to school. Describe all of the different towers that you can build
that are three cubes tall, when you have two colors available to work
with. Why were you sure that you
had made every possible tower and had not left any out? Indicate how your procedure can be
used to build towers that are four cubes tall, when you have two colors
available to choose from. 
Read Kieran &
Guzm¬án (2005), write an abstract, a reflection paper, and answer Week 4 DQR. 
Week 4 
Discrete and continuous
situations; using data to formulate an equation; linear relations 
TI73 Explorer 

Tasks: (1) Towers Problem
(3tall, 3colors) 
Discuss reading and
DQR Week 4 Task: Write a laboratory
report based on your investigations of the problems: "Temperature" and "A Big
Moosetake" 
Read Asimov (1982) and
complete the worksheet, ˆíReading Folklore for Mathematical Information.ˆì 
Read Zevenbergen
(2000), write an abstract, a reflection paper, and answer Week 5 DQR 
Week 5 
Scientific notation; using
data to formulate an equation; linear relations 
TinkerPlots TI73 Explorer 

Tasks: (1) Factors and
Number Properties and (2) Wai Muiˆïs quarter collection 
Performance assessment
and rubric scoring 
p is a factor of q, p is
a divisor of q, q is a multiple of p, and q
is divisible by q. 
Based on your
modifications of a plot of the first one hundred counting numbers, describe
as fully as you can patterns that you notice among each of the following sets
of numbers: square, triangular, and prime. Based on your pattern, determine what are the next five
elements greater than 100 of each set of numbers. 
Reread Zevenbergen
(2000) and revise abstract, reflection paper, and DQR answers. Read Powell (1993), write an
abstract, a reflection paper, and DQR. 
Weeks 68:Geometry 

Week 6 
Dynamic Geometry and
Exploring Triangle Centers 
The Geometerˆïs Sketchpad 

Discuss reading and
DQR Week 5 
Tasks: (1) Explore the
shapes created when the midpoints of the sides of quadrilaterals are
connected and determine whether there are any invariants or other
relationships. Based on your
exploration, develop a conjecture.
(2) Explore the shapes created when the midpoints of the sides of
triangles are connected and determine whether there are any invariants or
other relationships. Based on
your exploration, develop a conjecture. 
Read Peressini &
Knuth (2005) write an abstract and a reflection paper as well as answer week
7 DQR. 
Week 7 
Construction of geometric
figures and Exploring Properties of Quadrilaterals 
The Geometerˆïs Sketchpad 

Tasks: (1) How many
ways can you come up with to construct a rhombus? Try methods that use the Construct menu, the Transform menu,
or combinations of both.
Consider how you might use diagonals. Use the drag test on each construction. Write a description of each
construction method along with the properties of rhombuses that make that
method work. (2) Do the same for trapezoids. 
Read McGraw &
Grant (2005) write an abstract and a reflection paper as well as answer week
8 DQR. 
Week 8 
Exploring and Generalizing
the Pythagorean Theorem 
The Geometerˆïs Sketchpad 

Task: Create a visual
demonstration of the Pythagorean Theorem. 
Read Bakker &
Frederickson (2005) and write an abstract and a reflection paper. 
Weeks 912: Data Analysis and Probability 

Week 9 
Comparing related sets of
data; differences between categorical and numerical data; display data as
graphs. 
TinkerPlots 

Watch TinkerPlots
Basics; Tasks: Who Has the Heaviest Backpack; Is Your Backpack Too Heavy for
You? 
As a health and safety
expert, write a memorandum to your townˆïs Board of Education that discusses
your findings about students carrying backpacks that are too heavy (more than
15% of their body weight).
Include what percentage of students in the lower grades (1 and 3) and
in the higher grades (5 and 7) carry backpacks that are too heavy. Explain which students tend to carry
backpacks that weigh more for their body weight. Include graphs and explain how your graphs substantiate
your conclusions. 
Read Shaughnessy
(2003) and write an abstract and a reflection paper. 
Week 10 
Determining trends in time
series data; representing data to examine connections between two attributes 
TinkerPlots &
Probability Explorer 

Tasks: Menˆïs 100Meter
Dash at the Olympics; Men and Women at the Olympics 
Write a report for a
sports magazine in which you compare the records for men and women in the
goldmedal times for the 100and 200meter races as well as the goldmedal
distances for the high and longjump competitions. Determine whether the results show that one gender trends
to have better time than the other and indicate about how many seconds better
is the faster gender. In your magazine
report, use graphs to explain and justify your findings. 
Week 11 
Probability simulations;
Sampling with replacement, distribution, variance, and making inferences from
data 
Probability Explorer 

Tasks: Ten Marbles in
a Bag (physical enactment and calculator simulation) 100 Marbles in a Bag
(computer simulation). 
Write an opinion
editorial to your townˆïs newspaper that details your prediction of the
distribution of different marbles in the bag. Justify your prediction with representations of your data.

Read Jones, Thornton,
Langrall, and Tarr (1999) and write an abstract and a reflection paper, 
Week 12 
Sampling with replacement,
distribution, variance, and making inferences from data. Using and creating probability
experiments 
Probability Explorer & TinkerPlots 

Read Forman (2003) and
write an abstract and a reflection paper. Task: Schoolopoly 
Create a poster
exhibit for your court testimony that assets whether the dice sent you by a
manufacture is fair or biased.
Your poster must also illustrate compelling evidence that supports
your assertion. 
View video clips of 4^{th}
and 6^{th} graders working with Probability Explorer on 100 Marbles
in a Bag Problem and write a onepage description of the mathematical ideas
you observe that they engaged in. 
Weeks 13 14: Presentation and Evaluation 

Week 13 
Technology and Pedagogy
Test 


Read Tang and Ginsburg
(1999) and write an abstract and a reflection paper. 
Week 14 
Project Showcase Archiving Materials onto CD 


Explore
and analyze technologies available to learn and teach school mathematics in the
areas of number and algebra, geometry, as well as data analysis and
probability. Through class
discussions, problemsolving sessions, readings, writing assignments, student
presentations, and team projects, students will deepen and expand their
understanding of mathematics in addition to broaden their knowledge of methods
and materials for teaching mathematics, particularly in urban kindergarten
through 8^{th}grade classrooms.
As with all methods courses in the education sequence, students take
this course after their foundations courses and after being formally admitted
to the teacher education program.
A
Guide to Writing Reaction Papers
Mathematics
and Instructional Technology (21:300:348)
Arthur B. Powell
What is a reaction paper?
The
term, ˆíreaction paper,ˆì has many meanings, each depending on how an instructor
defines it. In this course, a
reaction paper is just what its name suggests—a piece of writing in which
you react to some text in a
reading—an article or a book chapter—by relating your experience or
thinking to the text. Your reaction may involve judgment or evaluation of a
main or subsidiary point of a reading; it may be an analysis of a point; or it
may raise a question triggered by a passage in the reading. In the paper, you will present a
brief, personal reaction rather than
a summary of the reading or attempt to provide either definitive judgments or
detailed analysis.
Specifically,
for each reaction paper, you will select two to three striking pieces of text
from the reading and write briefly about your reaction to each. You should interpret the word text
broadly. It can be a word, a
phrase, a sentence, or a collection of sentences; it can also be a geometric
figure, a mathematical expression, a table of values, or an illustration of
children using technology to do mathematics. The text you chose will be something that you find striking and, on the topic of the text, your
reaction will connect it to your experiential and intellectual life.
Think
of your reaction papers as serving three purposes: (1) a record of what you
found thought provoking in the reading, (2) an opportunity for you to deepen
and extend your pedagogical insights into the teaching of mathematics with
technology, and (3) an invitation for you to develop further your personal
philosophy of education.
A Reaction Paper
Qualities of an Interesting Reaction Paper
For this course, include at the top of your reaction paper your
abstract with the reference information of the reading and submit your paper to
the Digital Dropbox through Blackboard.
A
Guide to Writing an Abstract
Mathematics
and Instructional Technology (21:300:348)
Arthur B. Powell
What is an abstract?
Broadly
speaking, there are two distinct types of abstracts: descriptive and
informative.
Each is a short summary of a longer
piece of writing that highlights the major points covered and concisely
describes the content as well as the scope of the writing in abbreviated form.
You
are to write descriptive abstracts of course readings (articles and book
chapters). Think of your abstracts
as serving two purposes: (1) a useful study guide that you will read to recall
the important features and points of the reading and (2) a brief text that
professionals might read to decide whether it is worthwhile to read the full
article or chapter.
A Descriptive Abstract
Qualities of an Effective Abstract
Four Steps for Writing Effective Abstracts
1.
Reread the
article, paper, or report with the goal of abstracting in mind.
2.
After
you've finished rereading, write a rough draft without looking back at what you're abstracting.
3.
Revise
your rough draft:
4.
Print your
final copy and read it again to catch any remaining glitches.
Finally, for this course, include before your abstract reference
information for the reading in APA format, as in the sample below, and then
submit your abstract to Blackboard.
A Sample Abstract
Gopen,
G. D., & Smith, D. A. (1989). Whatˆïs an assignment like you doing in a
course like this? Writing to learn mathematics. In P. Connolly & T. Vilardi
(Eds.), Writing to learn mathematics and science (pp. 209229). New York:
Teachers College Press.
In
the context of a computerbased calculus course, the authors describe how
studentsˆï lab reports become learning tools. After instructors comment on the first drafts, students
revise their reports, applying principles of ˆíreader expectation,ˆì a writing
approach that they learn in a universityrequired writing course. By providing examples of student
writings, the authors show that revising forces students to recast and refine
their ideas into clearer and more precise form and, thereby, enables them to
deepen their understanding of the mathematical concepts and techniques of the
course.