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= Teaching Mathematics With Technology =

Technology can be used as a tool to do mathematics, medium to teach mathematics, or resource to plan for mathematics instruction. It is hard to know when it is appropriate to use technology for anyone of these three uses. The decision to implement a new technology is guided by the way we teach or intend to teach mathematics. Before 1990 the decision whether to use a calculator or computer in an activity depended on whether it would improve your students’ achievement on established objectives. “Will the technology help us do a better job of what we are trying to do?” The present era of mathematics education reform has complicated this instructional issue by advocating a more problematic than computational approach to mathematics instruction. We must decide what performances are indicators of good mathematical thinking before we can judge the possibilities of instructional tools.

Technology is Changing Mathematics
The ever changing uses of new technologies is one of the reasons the mathematics curriculum must change. Technology alters the nature of what is important to learn. In 1989 the NCTM Standards envisioned school mathematics as the “fundamental mathematics students will need, not just on the technological training that will facilitate the use of that mathematics.” Technology is part of the answer and its uses are promising even though we cannot completely or accurately predict or describe its eventual impact.If we really want to enable our students to succeed in life we need to prepare them to use, understand, control, and modify a class of technology that does not yet exist: we must teach a dynamical way of thinking mathematical. This does not necessarily mean use more technology in mathematics classrooms. At times this is counterproductive, in an effort to make mathematics accessible and attractive to a large number of students, good mathematical thinking is suffering (Cuoco, 1995). The mathematical thought processes needs to prepare our students to solve prospective problem situations using a whole new class of technologies.

** What is Appropriate Use of Technology When Teaching Mathematics **
How can we predict whether a new technology is going to encourage or short circuit mathematical thinking? One reason we educators have trouble predicting the instructional affect of different technologies is because we have not had time to understand how different technologies affect the procedural and conceptual knowledge of our students. The use of computers and graphing calculators many times dictates procedures to our students rather than enabling student directed procedures. As instructors we introduce students to mathematical procedures that reveal important concepts, but how soon or in what order should we introduce technology to explore all the mathematical variations?

===An instructional perspective to this dilemma is to ask, will this technological tool enhance the mathematical thinking of my students. To do this we must: ===
 * 1) Identify the mathematical thinking we expect the students will use to solve a problem.
 * 2) Determine whether technologies at our disposal will enhance their mathematical thinking or short circuit it.

===What does it mean to think mathematically? According to Kaput (1991) there are four types of mathematical thinking. ===
 * 1) Transformations within a representation--knowledge rules which guide actions on well used mathematical representations. Currently this type of activity, in the form of manipulation of symbols on papers dominates most mathematical activities.
 * 2) Translation between representations--express a mathematical notation or action in a multiple of representations: example-- graph y=2x + 4. In this example the student would need to translate between graphs and algebraic expressions.
 * 3) Construct a mathematical model from a representations within or outside of mathematics: Abstracting and modeling building are central to “doing mathematics”. A complex type of translation occurs when students are asked to translate between real-world situations and a mathematical structure or between mathematical structures.
 * 4) Consolidate relationships and/or processes into conceptual connections that organize mathematical ideas.

This last type of thinking is at the heart of the Standards--Students who are able to apply and translate among different representations of the same problem situation or of the same mathematical concept will have a powerful and flexible tool for problem solving.These four categories cumbersome definition for mathematical processes we teach every day. Hopefully a classroom examples of each of these types of mathematical thinking will bring the definitions to life. Use the attached document to identify the type of the mathematical thinking intended and the possible short-circuiting or enhancement of mathematical thinking.

Principles for Using Technology to Increase the Mathematical Rigor and Relevance of your Mathematics Instruction
Teaching mathematics with a high level of mathematical thinking (rigorous mathematics) and engagement (relevance) is difficult. Students resist challenging problems with no predescribed solution methods. Many students resist engaging in mathematics activities seem to have no connection to their lives or interests. It is the belief of the contributors of this wiki-page that technology can be an instructional solution for increasing the mathematical rigor and relevance of mathematical activities. Technology can can be used for students to check and explain their solutions. Mathematics solutions, processes, and explanations are not hidden in textbooks anymore but are readily available through technology. Technology can also be used for student to naturally communicate their understanding of the problems and solutions. The fact is, mathematics is now used to check the reliability of the information we find through technology. Student and teachers can focus on proficiency and accuracy of a smaller set of mathematics skills because technology empowers student to analyze a larger set of messy real world problems. Technology allows student the liberty to focus on the underlying concepts because technology can reveal what remains the same under change. <span style="color: #0b600b; font-family: 'Arial Black',Gadget,sans-serif;">Web 2.0 is a set of interactive web tools that allows users the ability to create and communicate their ideas freely. <span style="color: #0b600b; font-family: 'Arial Black',Gadget,sans-serif;">Visualization is a feature now accessible to all through technology and it allows users to see and analyze mathematical structures that were otherwise hidden.

===<span style="color: #0b600b; font-family: 'Arial Black',Gadget,sans-serif;">**Web 2.0 Tools** are an example of the new types of technology media that can be used to make mathematics instruction more rigorous and relevant. ===


 * <span style="font-family: 'Arial Black',Gadget,sans-serif;">Making Basic Facts Fun uses the web app Broken Calculator to teach and practice math basic skills. I have many ideas for teaching problem solving but I have trouble making the practices of basic math skills engaging and meaningful.


 * <span style="font-family: 'Arial Black',Gadget,sans-serif;">Picture Slopes uses a wikispace as an aid in finding the slope of a line given two points. The activity is unique in that students get to choose the lines they are going to find the slope of by picking their own picture to upload to the wikispace. This activity is educational because of the content, engaging because of the choice involved, and fun because students get to learn about each other and their interests through the pictures.


 * <span style="font-family: 'Arial Black',Gadget,sans-serif;">Exploration of Angles with iMovie uses iMovie as a tool for students to create a movie about finding angles of triangles using Trigonometry. Using iMovie naturally makes learning the concepts relevant to the stu<span style="font-family: 'Arial Black',Gadget,sans-serif;">dents because it allows students to be more creative which engages students in the activity. This lesson also ac**tivities their problem** **solving because the students have to determine a strategy to solving for the angle measures.**


 * <span style="font-family: 'Arial Black',Gadget,sans-serif;">**Learning Geometry through Billiards uses a video camera and a blog as tools to allow students to explore the geometry in billiards and a way to represent their solutions in a more technologically advanced generation. This lesson engages students by putting them in the math and challenges them to come up with a solution of how to hit a certain billiard ball into a pocket, with one major requirement: the cue ball must first strike a rail before sending this billiard ball into the pocket. This lesson gives the students a pool table layout with the scenario and asks them to go video themselves making the shot. After getting the video the students will take screenshots and use these to explain why the location on the rail they chose worked to bounce the cue ball off.**


 * <span style="font-family: 'Arial Black',Gadget,sans-serif;">Wonders of Weather uses ePals, or online pen pals from other countries to communicate mathematical data. The students from each country evaluate the weather where they are at. After that the students give surveys to determine if people's moods are dependent on the weather. The students then analyze and compare this data with their ePals to form their own statistical graphs and see if weather and mood are positively correlated. This lesson also includes a link that can be used to communicate with math teachers around the world in order to make ePal arrangements if you wish to do so.


 * <span style="font-family: 'Arial Black',Gadget,sans-serif;">Treasure Hunt uses edublogs as a place for students to post their group project. For this activity, the students are to first create and set-up an edublog account using edublogs.org as described in the full description of this project. Next, the students a to pick an object on the school's campus and create a treasure map with a complete list of mathematical clues that follow a rigorous set of guidelines. Once completed, the students are to post this project on their blog for other groups and classes to see. The way in which each group will be graded will be partly based on how well they are able to use their problem-solving skills to find another group's treasure and partly based on how well another group was able to find their treasure. For full instructions regarding this project, click on the link at the beginning of this project introduction.


 * <span style="font-family: 'Arial Black',Gadget,sans-serif;">Transformations of graphs uses Geogebra to help the students visualize graphs transformations. The students can work with the parameters of the graph to see how the function changes the algebraic representation or change the function to see how it changes the graph.


 * <span style="font-family: 'Arial Black',Gadget,sans-serif;">Why incorporate Web 2.0 tools in a Geometry class?. Gives some ideas on which web 2.0 tools to use in the classroom to make the lessons more rigorous and relevant. It contains an example of how to use Geogebra to work with special triangles.

<span style="color: #0d541c; font-family: 'Arial Black',Gadget,sans-serif;">Visualization is a feature of technology that can be used to make mathematics instruction more rigorous and relevant.

 * <span style="font-family: 'Arial Black',Gadget,sans-serif;">Making Visualization Activity Sheets uses the web app Dynamic Paper as a teaching resource to create hands-on worksheets for almost any mathematics lesson. The web app is found on the NCTM Illuminations website and easily be used to customize and print mathematics worksheets.


 * <span style="font-family: 'Arial Black',Gadget,sans-serif;">**Discovering Slope Intercept Form uses an on-line graphing calculator as an aid in discovering how changing parameters of an equation affects the graph of its line. The colorful on-line calculator allows students to graph more than one equation at a time. The activity contains a worksheet that guides students through the process asking them to analyze the similarities and differences of the equations along the way. This engaging activity would be useful as an introduction to graphing using slope intercept form.**


 * <span style="font-family: 'Arial Black',Gadget,sans-serif;">Discovering Polygon Interior Angle Sum uses the online program, Geogebra, to help students discover the formula relating the number of sides of a po<span style="font-family: 'Arial Black',Gadget,sans-serif;">lygon to it's interior angle sum. This lesson is engaging for students because they have the freedom to choose their own strategy to solving the problem, rather than simply being told what to do. This is also what makes this activity challenging and a problem solving lesson.


 * <span style="font-family: 'Arial Black',Gadget,sans-serif;"><span style="color: #0000ff; font-family: 'Arial Black',Gadget,sans-serif;">Radius of a Pipe uses an interactive math software program, Geometer's Sketchpad (GSP), to help students develop a relationship between two measureable distances related to a pipe (one being the radius). This lesson can be very relevant by bringing the pysical problem into the class. By bringing in a few pipes and representing the problem to them visually this will make the lesson the most relevant. After having the students walk through the steps towards visualizing this relationship, they will be challenged to apply this relationship to any pipe with a given radius. Also, this lesson can be made more challenging by having the students find another approach to generating the equation that GSP gives for the relationship between these two distances.
 * <span style="font-family: 'Arial Black',Gadget,sans-serif;">Modeling a Pendulum uses logger pro and a CBR to help students visualize how motion is represented in sign waves. The students are required to move the pendulum and evaluate it's movements to determine how the vertical and horizontal movements affect the amplitude and period of a sine graph. It is difficulut to simply imagine how these movements would affect the graphs especially without knowledge of calculus. If the students use logger pro and the CBR they can do this more effectively.


 * <span style="font-family: 'Arial Black',Gadget,sans-serif;">Historical Money uses WolframAlpha to help students interpret the data that is presented in graphs. The students are go to wolframalpha.com and type Historical Money in the search bar on the page. Once at that page, the students are asked to compare money values across time and interpret the direct meanings of the graphs presented by WolframAlpha. To make this activity more challenging and to check for understanding, the students are asked to compare and interpret the additional years that were provided in the given graphs. By using wolframalpha.com for this lesson, the students are able to quickly get to the task of doing several graph interpretations in a small or limited amount of time. This is a quick lesson and can be done easily as an entry task or as an exit task. For more information on this lesson, click on the link at the beginning of this introduction.
 * <span style="font-family: 'Arial Black',Gadget,sans-serif;">Special Triangles This gives some examples on how to use visuals to help the student connect mathematical concepts with real life. The page also contains a link to a web page where students can practice problems with triangles while at the same time having fun.

=<span style="color: #0c296f; font-family: 'Arial Black',Gadget,sans-serif; font-size: 18px;">Share Your Ideas = <span style="font-family: 'Arial Black',Gadget,sans-serif;">This wikispace is dedicated to improving mathematics teaching for beginning teachers. It is the goal of the editor that beginning teachers will learn how to take responsibility for student learning through assessing their ability to effectively use technology to enhance mathematical thinking and empower students to become engaged in meaningful mathematics. Beginning teachers will post mathematical lessons plans that they have created and explain how technology is used in their lesson.

=<span style="color: #153279; font-family: 'Arial Black',Gadget,sans-serif; font-size: 110%;">Activity Examples =

<span style="font-family: 'Arial Black',Gadget,sans-serif;">
<span style="font-family: 'Arial Black',Gadget,sans-serif;">The purpose of this lesson is to help students understand how linear regression can be used to make predictions with their own collected data.

=<span style="font-family: 'Arial Black',Gadget,sans-serif;">Investigative Mathematics =

<span style="font-family: 'Arial Black',Gadget,sans-serif;">The purpose of this unit is to engage students in an activity where they apply their mathematical knowledge of linear functions to real-world situations. The unit uses the computer program Logger Pro, iMovie, and digital cameras.

<span style="font-family: 'Arial Black',Gadget,sans-serif;">Graphing Equation
<span style="font-family: 'Arial Black',Gadget,sans-serif;">This lesson is a great use of calculators. It allows students to first use prior knowledge as to what a graph looks like. Then it goes into how to use a calculator for graphing equations. Finally ending with students doing multiple graphs and equations. This lesson explores the advancement of difficult equations.

<span style="font-family: 'Arial Black',Gadget,sans-serif;">Calculating the Friction Coeficient
<span style="font-family: 'Arial Black',Gadget,sans-serif;">This lesson is a good lab exercise for interpretation of data. The students collect data on a computer and analyze the plot. The lesson uses "line of best fit" and integrates mathematical reasoning with technology.

<span style="font-family: 'Arial Black',Gadget,sans-serif;">Slinky Graphs

This lesson uses a Calculator Based Ranger (CBR) to encourage students to investigate the sine function.
<span style="font-family: 'Arial Black',Gadget,sans-serif;">Lesson Plan Linear Regression Statistics-The Hand Squeeze

<span style="font-family: 'Arial Black',Gadget,sans-serif;">The purpose of this lesson is to give students a better understanding of the slope of a line. Using technology, the students will better understand slope and its relation to reaction time.

<span style="font-family: 'Arial Black',Gadget,sans-serif;">Exploration with Pythagorean Theorem

<span style="font-family: 'Arial Black',Gadget,sans-serif;">This lesson gives students experiences in applying the Pythagorean Theorem and it's converse.

<span style="font-family: 'Arial Black',Gadget,sans-serif;">Parallel and Perpendicular Lines

<span style="font-family: 'Arial Black',Gadget,sans-serif;">The purpose of this unit is to familiarize students with the applications of properties of parallel and perpendicular lines using mathematical technology.

<span style="color: #cc1eeb; font-family: 'Arial Black',Gadget,sans-serif;">Tsunamis and Sine Waves
<span style="font-family: 'Arial Black',Gadget,sans-serif;">Students will use their knowledge of sine waves to explore connections to the world such as tsunami waves. Students will analyze the tsunami wave at different times as it approaches shore. They will use a program called Logger Pro to translate the wave to a sine equation.

<span style="font-family: 'Arial Black',Gadget,sans-serif;">Bank On It

<span style="font-family: 'Arial Black',Gadget,sans-serif;">During this lesson students will review SOHCAHTOA and what it means. Also, students will learn how to apply and find lengths of right triangles using the sine, cosine, and tangent ratios. Finally, we will briefly talk about Pythagorean's Theorem, and special properties of 45-45-right triangles, but students will not be tested on this. This lesson is strictly for sine, cosine, and tangent.

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