Technology in Mathematics

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Introduction

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Quipu, circa 1500
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Mathematica, circa 2000

The field of mathematics has benefited from technology throughout its history. Mathematical tools have advanced from the abacus and Quipu, an Incan base-10 counting system made of knotted fibers, to the calculators and computers of today. These tools can be used in the classroom to promote higher-level thinking and highlight the links between taught concepts and their real-world applications.

The range of instructional technology applicable to mathematics is vast, and at times, overwhelming. A multitude of hardware, software, and online offerings can be implemented in both conventional and novel ways to complement the many content, process, and technological standards of successful math instruction.

by Alex.ragon

Rationale

Many benefits provide a sound Rationale for Technology in Education, including relative advantage, increased accessibility, cooperative grouping, exposure to new technology, interactivity, and differentiation.

In math, students can exercise higher order conceptualization to focus on "decision making, reflection, reasoning, and problem solving" (National Council of Teachers of Mathematics), such as by manipulating the variables of an equation and observing its effects on a graphing calculator. Instead of spending a significant amount of classroom time on computation, students can focus on higher level skills and useful applications of their newfound knowledge.

Standards

NCTM Standards for School Mathematics

The National Council of Teachers of Mathematics (NCTM) places great emphasis on the use of technology in mathematics education. Technology, one of the six principles fundamental to school mathematics programs as outlined by NCTM's Principles and Standards, is "essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning."

In the NCTM Standards for School Mathematics, NCTM outlines ten knowledge and skill objectives that should be achieved from prekindergarten through grade 12, including Numbers & Operations, Algebra, Geometry, Measurement, Data Analysis & Probability, Problem Solving, Reasoning & Proof, Communication, Connections, and Representation.

ISTE National Educational Technology Standards

The International Society for Technology in Education (ISTE) publishes the 'National Educational Technology Standards (NETS)', a set of guidelines for the successful use of technology in education. Teachers utilizing technology in their instruction should assess their implementation by referring to NETS, detailed in the Rationale for Technology in Education.

Issues

Despite the many benefits of technology in math instruction, Issues and Barriers to Integrating Technology exist that must be addressed. The greatest obstacles to technology in any field of education are funding and access. Teachers and students need access to the hardware and software before any useful application of the technology can occur, and teachers require professional development and construction time to utilize technology effectively.

Fortunately, the importance of technology as part of the classroom and beyond is increasingly recognized, and attention is being paid by school districts and educational organizations to the issues that stand in the way of technology in education. The following are issues specific to mathematics classrooms.

Calculator Dependency

Integrating technology into mathematics is not without controversy. Many educators fear that calculators and other technology will eliminate opportunities for students to sharpen their computational abilities. Teachers and students alike must use calculators as a tool rather than a crutch, as a means of extending math in new forms, exercising higher order thinking skills, and working more efficiently and accurately, without sacrificing an underlying comprehension of the concepts.

Access

Technology is increasingly becoming an critical component of advanced high school math courses. E.g. students are expected to be proficient in the use of graphing calculators to solve problems efficiently. Additionally, calculator use is permitted on standardized tests such as the SAT and AP exams. As graphing calculators and other technologies become an integral part of the math classroom, care must be taken to provide all students with equal access to these important tools.

Mathematical Input

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Microsoft Equation Writer

More than other subjects, mathematical input on a computer poses a challenge. Teachers must oftentimes compose complex mathematical expressions which are nontrivial on a word processor or presentation program, such as fractions and radicals. Fortunately, several solutions exist, from commercial WYSIWIG software that integrates with Microsoft Word (Design Science MathType) and free markup software (LaTeX) to programs that accept handwritten input on tablet PC's and converts it to typewritten images (Microsoft Equation Writer). Additionally, graphing calculator adapters/links offer a method of displaying calculator output and graphs easily on computer and projector displays.

Effective Uses of Technology in Mathematics Instruction

Graphing Calculator Lab

The Texas Instruments Calculator-Based Ranger (TI CBR) and other data collection devices exist that can interface with standard graphing calculators. The data collection capability allows students to experiment, collect real-world data, and fit it with the mathematical concepts they have learned in the classroom.

View a library of graphing calculatory lab activities at the TI Activities Exchange and the example below. There is also a TI Emulator that allows your computer screen to mimic the screen of a TI calculator.

Activate the CBR and bounce a racquetball beneath it. The CBR will record the velocity of the ball over time, and send it to the calculator to be graphed. From there, students can apply the graphing calculator functions to find the best fit quadratic function. The following questions can be posed.
1. Describe the characteristics of the velocity of the ball through its trajectory from one bounce to another.
2. Where is the ball when the velocity is zero?
3. The standard equation for the position of a projectile is $s = s_{0} + v_{0}t + \frac{1}{2}at^{2}$. What is the quadratic equation that models the position of the ball over time?
4. Using this equation, identify the initial height $s_{0}$, the initial velocity $v_{0}$, and the acceleration $a$.

WebQuests

A WebQuest, as defined by creator Bernie Dodge, is "an inquiry-oriented activity in which some or all of the information that learners interact with comes from resources on the internet." WebQuests engage students in meaningful activities and have them apply new concepts to areas they find relevant.

View an example below, or choose from a wide selection of Best WebQuests or for math students in K-2, 3-5, 6-8, and 9-12.

Following a unit on linear equations and with a primer on piecewise functions, students can go on the following webquest to analyze the cost-effectiveness of several real cellphone companies.
1. Navigate to the Cingular, Sprint, T-Mobile, and Verizon websites. Select the most basic personal plan that is offered and create a table that makes note of the monthly cost, included minutes, and cost per additional minute for each.
2. For each company, construct a piecewise function that represents monthly cost as a function of the number of minutes.
3. Graph the functions on a graph, with the number of minutes on the x-axis and the monthly cost on the y-axis.
4. Consider the following questions and write an essay explaining your findings. Include your data table and graphs, in addition to any additional supporting evidence.
What does it mean when a company's graph is above another's?
What occurs at the point where two companies' graphs intersect?
Is one single company the most cost effective overall or does it depend on the number of minutes used per month?
Examine a sample cellphone bill, either provided by your parent or the teacher. How many minutes were used in the past month?
Compute the cost under the basic plan of each cellphone company. Which is the most cost-effective?
What factors besides cost can be considered when choosing a cellphone company?

Presentation Technologies

Randy from EDTECH shares that the must-have technologies in his classroom start with presentation hardware and software, including a laptop, a digital document camera, a presentation system, mimio whiteboard, TI SmartView graphing calculator emulator, and Geometer's Sketchpad. View a video of some of these technologies in action at Annenberg Media Learning Math.

Distance Learning

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WorldWideWhiteboard

Distance learning may be the best solution for students in a variety of circumstances, including those in rural areas, interested in remediation or acceleration, home-schoolers, and students whose home lives prevent them from attending school. With an internet connection, students have a link to the outside world from their computers, with which they can attend online lectures, complete assignments, and communicate with tutors via online whiteboard.

A number of universities and other organizations offer distance math courses for students in K through 12, including the Center for Talented Youth at Johns Hopkins University, NetMath at the University of Illinois, the Education Program for Gifted Youth at Stanford University, and Larson Math.

Real World Problem Solving

Math students can be tasked with solving problems relevant to their school or community. Art from the EDTECH mailing list provides examples of how when the school district was searching for location for a new bus garage and retrofitting a school with air conditioning, students used contractor information, mapping websites, drawing programs, geometry, and trigonometry to find and support their own solutions.

Emerging Technologies

Emerging Technologies such as Weblogs, Wikis, and Podcasts can be utilized with great effectiveness in a math classroom. Darren Kuropatwa's blog A Difference chronicles his Applied Math, Precalculus, and AP Calculus classes as they participate in reflective writing, homework collaboration, textbook authoring, online classes, and podcast publishing.

Hardware

Other

• Document Cameras High resolution cameras designed to present detailed documents and images to a classroom
• mimio Whiteboards Control desktop applications and documents directly from the board, and digitally capture notes and drawings that can be saved and shared
• Tablet Computers The combination of touch screen hardware and optical character recognition software offers effortless input of mathematical expressions with all of the benefits of electronic media such as saving and sharing of data
• TI-Calculator-Based Laboratory (CBL) Integrates with TI calculators and up to 40 probes that measure temperature, voltage, light, and more
• TI-Navigator Creates a wireless network of TI graphing calculators that allows for collaborative work, instant assessments and polls, screen monitoring, and data sharing
• TI-SmartView TI-84 Plus emulator software that can be used as a demonstration tool for leading the classroom instruction of math and science concepts

Software

Links to websites for math software

• Cognitive Tutor This Carnegie Learning product provides software-based, individualized computer lessons to students of algebra I, geometry, and algebra II
• Mighty Math Cosmic Geometry
• Descartes' Cove A Johns Hopkins University designed exploratory game for grades 6 to 8 math students
• Enriched Math A strategic game designed to get students in grades 5 through 8 talking about math
• Exploring Math Computer activities designed to help students visualize math and strengthen problem solving with a focus on number and operation, algebra, geometry, and probability
• GAMCO Colorful education software appropriate for math students from grades K through 9
• MathMedia Instructional math programs in arithmetic, algebra, geometry, trigonometry, and precalculus
• Math Blaster A fast action math adventure for ages 6 to 12
• Math Games A library of math games maintained by USD 298 for students of all ages
• Mighty Math Zoo Zillions, Millie's Math House, and Carnival Countdown introduce addition, subtraction, story problems, number facts, 3D shapes, counting money, and making change for grades K through 2
• National Library of Virtual Manipulatives A Utah State University created gallery of virtual manipulatives
• Plato Learning A variety of instructional and exploratory programs for arithmetic all the way through calculus 2
• Riverdeep In addition to several enrichment programs, instructional offerings range from pre-primary to algebra and geometry
• Tom Snyder Productions Creators of Fizz and Martina's Math, PrimeTime Math, and many more computer programs appropriate for K through 12

Math Software Tools

• Design Science MathType equation editor, WebEQ interactive math on the web, MathFlow math publishing, and MathPlayer MathML browser plug-in
• GCalc Online Java graphing calculator
• Geometer's Sketchpad A mathematics visualization tool with mathematical notation and graphing functions suitable for geometry, algebra, trigonometry, and calculus. This program is great for visualizing concepts that are sometimes hard to understand.
• Cabri - Geometric figures construction & experimentation
• Maple Mathematical productivity tool capable of numeric, symbolic, and graphical operations
• Mathematica A symbolically programmable calculator with data, function, and graphical operations
• MathJournal An interactive program for the Tablet PC that provides a natural and intuitive environment for solving mathematical and engineering problems
• Microsoft Education Pack Five free programs for educators to maximize productivity on their tablet PC's, including Equation Writer which translates hand written expressions into typed ones.
• Sage is an open-source alternative to Mathematica, Maple, etc.
• WorldWideWhiteboard An interactive graphical chat tool designed to allow realtime communication of mathematical and graphical concepts.
• Upper School software: Geometer's Sketchpad, Wolfram Mathematica (super smart math students), Simcalc Mathworlds, AutoGraph, Derive, True Basic

Web Resources

For Students

• AAA Math Explanations for hundreds of math skills with interactive practice pages
• Ask Dr. Math An archive of math questions submitted by students and answered by Dr. Math
• Math Buddy Online Hundreds of interactive activities and manipulatives, interactive practice worksheets, and assessment modules focused on math concepts for grade 2 to grade 5 math.
• Create A Graph An online tool for creating bar, line, pie, and x-y graphs
• Fraction Fun Fun interactive animations on the basics of fractions
• MathWorld A comprehensive and interactive mathematics encyclopedia
• Mighty Math Club A website for kids that's devoted to math fun and learning
• Numerical Mathematical Utilities Several numerical mathematical utilities. For example, utilities for solving the Quadratic, Cubic, and Quartic Equations; solving N Equations in N Unknowns; Eigenvalues and Eigenvectors; and more.
• Purplemath Online algebra resource
• Visual Calculus A collection of interactive modules that can be used in the studying of calculus
• Wisweb Hours of interactive, addicting math games; the geometry games are outstanding; be advised, it's a Dutch translation
• Shodor Interactive Several interactive (java-based) activities for a variety of levels and topic areas.

Assessment

1. As the sun is eclipsed by the moon, the batteries in 29 of 30 graphing calculators go dead. How can technology be used to maximize the effectiveness of the lone functioning graphing calculator in the classroom?
2. Inputting mathematical expressions into computers can oftentimes be as unwieldy as picking a lock with a peanut. Do solutions exist that can make the process more facile?
3. What are the major issues of concern when it comes to technology in the math classroom?
4. A student breaks his leg at math camp and is housebound for the remaining six weeks of the summer. How can technology be used to fulfill his enthusiasm for math?
5. Several of your students are organizing a Texas Instruments boycott. Can you provide them with alternative companies that manufacture graphing calculators?

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