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

Download Syllabus as a PDF

I. Overview

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

- subject-centered curricula in a teacher-centered 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 countryàÔs students. This focus has prompted pedagogical shifts away from concentrating on algorithms and computations toward emphasizing

- student-centered curricula with teacher as facilitator or coach;

- discovery activities, open-ended 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 studentsàÔ 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.

II. Objectives

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, problem-solving 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 economic-class imperatives for access to technology-rich, 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 technology-rich mathematics classrooms.

III. Relationship to Education Program

This course is part of the sequence of courses required for elementary-middle school New Jersey State certification.

IV. Two closely related courses

21.300.342 Elementary Mathematics and Pedagogy

21.300.343 Elementary Science

V. Pre- and co-requisites

By permission

VI. Learning Outcomes

By the end of this course, you should be able to perform the following skills with the listed technology:

Microsoft Word

1. Insert a M x N table
2. Insert an Equation using the Equation Editor that contains a fraction, radical, exponent, parentheses, and at least 3 mathematical operations.
3. Insert from another application a Screenshot or Picture that does NOT float over the document.
4. Insert 2-D and 3-D geometrical figures using the Drawing Tools
5. Create a Hyperlink to: 1) an Internet website, 2) another Word document, 3) a file from either Excel, Sketchpad, TinkerPlots, or Fathom.

TI-73 Explorer Calculator

1. Use Constant function with counter-creating 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)

1. Explore data sets, create graphs: scatter plots, time series graphs, histograms, and box plots.
2. Add a trend line to a scatter plot.
3. Enter Data in either a New Collection or a New Case Table with appropriate Attribute Names.
4. Import raw, case-based data or microdata from the Internet, using Import for URL, from the site http://lib.stat.cmu.edu/DASL/DataArchive.html into New Collection or a New Case Table.
5. Be able to Lock and Unlock a Collection
6. Enter data in Collection or Case Table based on a Formula that uses at least one built in Function
7. Create a Function Plot with a New Plot and Enter a Formula for the Function you wish to graph.
8. Add a New Slider

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 Least-Squares 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, à‚

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, à‚

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

1. Run a 1-event experiment with Coins or Dice.
2. Run any experiment over 1000 trials with appropriate displays of the data
3. Save Images of a Bar and Pie Graph in the Notebook and also be able to Copy and Paste these images into a Word document.
4. Change the probability in an experiment using the Weight Tool and Hide the Weight Tool with a Password.
5. Save an experiment and Open an existing file.
6. Design and Run a Marble experiment with at least 3 different colors and at least 12 marbles in the bag.
7. Use the RunUntil tool
8. Be able to Design your own experiment.
9. Use the MakeIt Tool to create all possible outcomes for the experiment.
10. Show the outcomes of a 2-event experiment both Ordered and Unordered in both the Data Table and the Stack Columns.

Microsoft Excel

1. Enter data in columns and rows
2. Import data from Internet website and format into columns
3. Format cells and text with Color and Font Size and, resize rows & columns, and add borders to cells
4. Perform basic operations (e.g., sum, average, random) and create formulas using Excel functions and mathematical operations with cell references
5. Create basic types of graphs with appropriate data, label axis, and change axis scale (e.g., Line graph, bar graph, pie graph, X-Y plot)
6. Insert a scroll bar and link to control a cell's value. Use the controlled cell in a formula that generates a table of data and/or a graph that will all update when the scrollbar is changed.
7. Add a trendline to an X-Y scatterplot and display formula and r2 on chart

VII. Materials and Reading List

1. Mathematics Education Bundle, published by Key Curriculum Press <http://www.keypress.com/>, a specially-priced, 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. 149-167). New York: Harper and Row.

5. Powell, A. B. (2001). Capturing, examining, and responding to mathematical thinking through writing. Pythagoras (55), 3-8.

6. Huinker, D. (2002). Calculators as learning tools for young children's exploration of number. Teaching Children Mathematics, 8(6), 316-321.

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.), Technology-supported mathematics learning environments (Vol. Sixty-Seventh 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. 356-369). 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. 201-223). 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.), Technology-supported mathematics learning environments (Sixty-Seventh Yearbook, pp. 277-290). 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.), Technology-supported mathematics learning environments (Vol. Sixty-Seventh Yearbook, pp. 303-317). 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.), Technology-supported mathematics learning environments (Vol. Sixty-Seventh 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 K-12: Readings from NCTM's school-based journals and other publications (pp. 232-237). 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. 151-178). 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. 216-226). 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 K-12, 1999 Yearbook of the National Council of Teachers of Mathematics (pp. 146-155). 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 K-12, 1999 Yearbook of the National Council of Teachers of Mathematics (pp. 45-61). Reston: National Council of Teachers of Mathematics.

20. National Library of Virtual Manipulatives for Interactive Mathematics: http://matti.usu.edu/nlvm/nav/index.html

VIII. Evaluation Criteria

Readings, problem solving, and written reflections 20%

Portfolio of selected, completed assignments 30%

Project and showcase presentation 30%

Final Examination 20%

IX. Week-by-week list of topics and tools as well as readings and assignments

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 classàÔs electronic list on Blackboard, distribute a brief assessment of your experience writing in connection with learning mathematics.
Weeks 2-5: 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 ChildàÔs 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	TI-73 Explorer	- Discuss reading and DQR - Tasks: (1) Five Steps to Zero and (2) Towers Problem (4-tall, 2-colors) - Problem-solving 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¨·n (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	TI-73 Explorer	- Tasks: (1) Towers Problem (3-tall, 3-colors) - 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, àÌReading Folklore for Mathematical Information.àÏ - 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 TI-73 Explorer	- Tasks: (1) Factors and Number Properties and (2) Wai MuiàÔs 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. - Re-read Zevenbergen (2000) and revise abstract, reflection paper, and DQR answers. Read Powell (1993), write an abstract, a reflection paper, and DQR.
Weeks 6-8:Geometry
Week 6	Dynamic Geometry and Exploring Triangle Centers	The GeometeràÔs 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 GeometeràÔs 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 GeometeràÔs Sketchpad	- Task: Create a visual demonstration of the Pythagorean Theorem. - Read Bakker & Frederickson (2005) and write an abstract and a reflection paper.
Weeks 9-12: 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 townàÔs 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: MenàÔs 100-Meter 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 gold-medal times for the 100-and 200-meter races as well as the gold-medal distances for the high- and long-jump 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 townàÔs 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,