Evaluation+and+Annotation

** Math Snacks-Pearl Diver **
This applet provides students with the opportunity to learn, practice, play, and explore fractions as part of a whole. Pearl Diver's promotes understanding of rational numbers. It is used to emphasis basic to more advanced fractions as part of a whole and it's value on a number line. The use of slices and fractional location on a number line promote relational understanding of fractions by providing students with a visual representation of each fraction as well as equivalent fractions on the number line. This applet will help students understand and see the connection between fractions and division as well.
 * What is being learned? What mathematics is the focus of the activity/technology? Is relational or instrumental understanding emphasized?**

Learning takes place when the user takes into consideration the number of parts in the whole number or numbers. Once the learner realized how many parts there are, they need to be able to correctly identify several different fractions on the number line in a short amount of time. After a student has shown mastery of a level, they are presented with a whole eel that needs to be cut into equal parts. A student is continually challenged with increasingly harder problems as they master each set of fractional skills.
 * How does learning take place? What are the underlying assumptions (explicit or implicit) about the nature of learning?**

This technology allows students to learn and practice the relationship between fractions and whole numbers in a fun, dynamic way. The applet provides instant feedback to the students when they are correct and when they make a mistake. Pearl Diver's also uses time as a factor to motivate students to accomplish their tasks quickly and accurately. The applet is unique in that it continuously challenges the player as the fractions get more challenging. One disadvantage of the game is that there is no explanation when a student gets a problem wrong. You are only shown a boot instead of a pearl. Another disadvantage of this technology is that it does not offer students different levels to work in based on skill or desire.
 * What role does technology play? What advantages or disadvantages does the technology hold for this role? What unique contribution does the technology make in facilitating learning?**

This applet is mostly meant to supplement instruction of rational numbers, which is already central to all curriculum's grade 3-8. Pearl Diver's is a perfect way to continue or enhance the learning that is going on in the classroom. However, I can see a student at younger ages being interested in its' game-like features and learning these skills independently of the classroom.
 * How does it fit within existing school curriculum? (e.g., is it intended to supplement or supplant existing curriculum? Is it intended to enhance the learning of something already central to the curriculum or some new set of understandings or competencies?)**

This technology do not interact with the social context of learning. One might be able to set challenges between students or classes, but this is not a feature of this applet.
 * How does the technology fit or interact with the social context of learning? (e.g., Are computers used by individuals or groups? Does the technology/activity support collaboration or individual work? What sorts of interaction does the technology facilitate or hinder?)**

Learner's differences are not taken into account. Each player starts at a basic level and is promoted to the next level when completed. A student is not able to enter at a different level based on ability or interest.
 * How are important differences among learners taken into account?**

It would be helpful for players to understand the basic concept of fractions in order to be successful in this game but it is not required. Teachers and students need to read directions carefully before beginning. Many students might want to just click play and begin without understanding how to complete a task. Once students understand that they must dive at the correct location, while missing the eel, they will quickly learn that getting the most points requires speed and accuracy. Innovative learning games like this will continue to challenge and excite students.
 * What do teachers and learners need to know? What demands are placed on teachers and other "users"? What knowledge is needed? What knowledge supports does the innovation provide (e.g., skills in using particular kinds of technology)?**

Fractional Model
This applet is focused on understanding the various forms of numbers. The visual representations allow students to better understand fractions and decimals. The fact that fractions, decimals, mixed numbers, and percents are all displayed allows students to understanding the connections between different types of numbers. There is even the option of storing different numbers in a table to compare them directly. Students can also use this resource to study by anticipating the various forms of a number prior to selecting a new numerator and denominator. By allowing the numerator and denominator to be manipulated, this applet definitely aims at understanding of positive rational numbers, such as fractions, whole numbers, and percents. This resource can also be used to better understand division. Besides addition, there are not any other operations taking place with the fractions directly. This resource targets relational understanding. The purpose is to help students gain a deep understanding of fractions, not to simply perform computations. Students are constructing deeper meaning through the manipulations and the visual representations. There is no direct instruction how to transition from decimals to percents, and to fractions. Still, students can utilize the table and store numbers to understand the relationships between different numbers. The table will allow students to make and test conjectures about the different types of numbers.
 * What is being learned? What mathematics is the focus of the activity/technology? Is relational or instrumental understanding emphasized?**

Technology allows for visual representations to be displayed so that students can better understand fractions. Technology allows for the manipulations of the numerator and denominator, which fosters the constructivist aspect of this resource. An instant calculation of fractions, decimals, percents, and mixed numbers is available for students due to the technology. Learning through exploration is at the forefront of this applet. Students explore the relationship between the numerator and denominator. Students also are able to explore the relationships between fractions, mixed numbers, percents, and decimals by manipulation. This could be used for direct instruction in conjunction with a smart board, but not as a stand alone resource. This could also be used as a component of practice if students were to transition from one type of number to another and then check their answers. However, the resource itself does not provide inherent practice.
 * What role does technology play? What instructional function(s) does the resource serve?**

Numbers are represented in symbolic form, as fractions, decimals, and percents. There are also visual representations which assist student understanding and are in the form real world objects. Graphical representations are also used with pie charts and with students being able to select the numerator and denominator off of the number line.
 * What kinds of representations of the mathematics are used?**

Math Baseball
The main focus of this resource is for students to perform computations. Students are able to select addition, subtraction, multiplication, or division for single digit mental computations. Also, the difficulty can be increased for problems that require pencil and paper. Problems can be presented in algebra fashion (as fill in the blanks) as well. Unfortunately, there is no timer which does not support rapid computation. This resource allows students to select either addition, subtraction, multiplication, or division as a topic to practice. Students then are able to select the difficulty, which determines if students will be given single digit whole numbers or two, three, or four digit whole numbers. This resource is aimed at skill and drill exercises. When the difficulty is increased students need to be able to carry, burrow, add place holders, and handle remainders. Still all of those tasks can be memorized without understanding as to why the rules work. Therefore, this understanding is instrumental.
 * What is being learned? What mathematics is the focus of the activity/technology? Is relational or instrumental understanding emphasized?**

Technology offers students a different approach to learning in the form of a game. Through technology students are able to compete in a game with external rewards for correct answers. The game allows for differentiated practice, as students are able to vary the difficulty. Additionally, the game provides students with immediate feedback on their progress. This applet allows students the opportunity to practice computational skills and speed with which they have already been exposed. Students can alter the difficulty to increase their ability to work with larger numbers. Students would not be able to learn addition, subtraction, multiplication, or division from this resource alone. This also does not promote higher level learning involving the connection between operations.
 * What role does technology play? What instructional function(s) does the resource serve?**

The type of representation being used is symbolic representation. Students are quizzed on addition (+), subtraction (-), multiplication (X) and division (÷). The symbolic form is structured in the traditional manner with the digits lined up for addition, subtraction, and multiplication. For division, the digits are lines up as well, requiring students to rewrite before dividing.
 * What kinds of representations of the mathematics are used?**

Dividing Fractions by Fractions
This application allows students to learn and practice the division of fractions. There are two sections: //Learn// and //Explore.//In the first section students are presented with a written explanation of how to divide fractions and with two examples. They are reminded to invert and multiply, to simplify the result and are presented with an example on how to simplify before multiplying. In the second section they have the opportunity to play and practice. The program randomly selects two fractions and students write the result of the division. Immediate feedback is provided.The program allows for a 60 seconds time limit, a setting where extra time is given for each correct answer, and a 20 questions race. Because the applet offers the option to solve a set of problems in a certain amount of time it reinforces fluency. The resource is focused on the division of fractions. Students also practice multiplication when multiplying with the reciprocal. The main focus of this resource is instrumental understanding. Students are expected to carry out procedures in order to get a correct answer. There is no explanation of why the division of fractions is equivalent to multiplication by the reciprocal and why it is possible to "cross cancel " and simplify before multiplying.
 * What is being learned? What mathematics is the focus of the activity/technology? Is relational or instrumental understanding emphasized?**

In this case technology supplements instruction and offers another context for practicing in order to increase fluency. This application is intended to be used to practice previously learned skills, but it could also be used for direct instruction because students are presented with the algorithm and with two examples prior practicing. The explanations are not enough, though, to promote relational understanding. Students are less likely to be able to solve the exercises correctly by just following the examples, without having prior knowledge of division of fractions.
 * What role does technology play? What instructional function(s) does the resource serve?**

Fractions are represented symbolically and there are also symbols for division.
 * What kinds of representations of the mathematics are used?**

Match the Fraction
This applet is used to emphasis basic fractions as part of a whole and it's value on a number line. The students are expected to divide the number line of 1 into slices and select the fraction that is presented. The use of slices and fractional bars promotes understanding fractions as division of a whole number. The denominator determines how many "slices" you break the whole into. The numerator determines part of the whole. This allows students to better visualize where fractions lie on the number line. However, this applet will not recognize equivalent fractions. If the students are presented with 2/4 and the student attempts to model 1/2 the program will not accept it as correct. Improper fractions are not included in the program, nor are whole numbers. This resource is intended to aid in the understanding of fractions mostly, but it does represent fractions as the division of the whole number 1 into slices. The focus of this applet is to create relational understanding, but fails to provide any great depth in understanding fractions. Therefore it is also instrumental because after solving a few exercises, they will automatically choose the right number of parts , and the number of parts they need to highlight without reflecting. Students are able to see and pick out the fractions on the number line as wells as parts of a whole, but not much else. The program does not recognize equivalent fractions. Therefore, students are not learning that 1/2=2/4 and are in fact led to believe that the two are not equal due to the fact that only one will be recognized as correct. Second, the number line can only be divide into equal parts. Students are not able to see how 4/5 relates to 2/3. Instead students only compare fractions with the same denominator.
 * What is being learned? What mathematics is the focus of the activity/technology? Is relational or instrumental understanding emphasized?**

The role of technology here is to supplement instruction by providing students with an interactive model that shows the students the fractional meanings as well as its place on a number line through visual representation. I think that it has the possibility of being a great resource if you were able to apply your understanding of equivalent fractions as well as bigger fractions and perhaps even mixed numbers. This applet would function as practice, direct instruction, and exploration. It provides a visual explanation and a different method of representing the material to students. It also allows student to understand visually that the denominator is the number of segments of a whole and numerator is the parts of the whole. This applet could be used to introduce the concept or follow instruction an other hands on methods that can be applied to learning fractions. This applet could also be used an introductory game to see if the students can understand the concept of fractions through their own efforts.
 * What role does technology play? What instructional function(s) does the resource serve?**

Numbers are represented in symbolic form, as fractions, decimals, and percents. There are also visual representations which assist student understanding and are in the form real world objects. Graphical representations are also used with pie charts and with students being able to select the numerator and denominator off of the number line.
 * What kinds of representations of the mathematics are used?**

Free Ride
Students were expected to understand numbers. They were comparing ratios of numbers to move the bike a certain distance. Students are also working with adding and subtracting fractions to help move the bike a certain distance. Students were suppose to learn about fractions and ratios. They were looking how the front gear would affect the back gear. I would say that the applet did a poor job of students understanding of fractions. The applet was easy to just guess which gear should go where. If students would be required to keep a table of different gears and how far the car went, the goal of the lesson could be achieved better. The applet forces students to make relational understanding. They are to make connections between how the bike will move and the ratio they are creating. Eventually the students will begin to apply certain steps to solve the problem, so there will be some instrumental understanding as well.
 * What is being learned? What mathematics is the focus of the activity/technology? Is relational or instrumental understanding emphasized?**

This task would be almost impossible to do if it were not for the technology. The technology provides the student with visual representation of fractions as well as shows students how a bike works. The students, if asked to create a table could learn ratios and fractions through exploration. It could also be used as a real world context for fractions and ratios. This applet also provides students with the opportunity to practice working with fractions and ratios.
 * What role does technology play? What instructional function(s) does the resource serve?**

The representation of the math was spatial as the students saw how the ratio between the two gears affected the distance the bike went. It was also a dynamic representation of math as students began to see the the distance the wheel turned was the distance the bike traveled.Students are also connecting ideas from a math classroom to a real world object, the bike. This applet focus is on representing the mathematics through the use of dynamic visual/spatial tools. The motion of the bike reinforces the concepts of ratios as fractions.
 * What kinds of representations of the mathematics are used?**

TinkerPlots: Is Your Backpack Too Heavy For You?
Students are learning to use a set of data to create graphs in order to compare the data. Students are investigating the amount of students that are carrying backpacks that are too heavy (more than 15% of their body weight). To answer this question and subsequent questions, students have to organize and analyze data based on percentages. Therefore, the focus of the activity is to look at data and percentages in order to come to conclusions about the student population. The understanding is more relational because students have to process the data and relate it to what they know about backpacks and students' weights to answer the questions. Thus they are connecting information to form a better understanding which is a part of instrumental understanding. In performing calculations of percentages, this could also be seen as instrumental understanding as well. For many students this will be a set process or formula used in the calculation. This activity uses the assumption that learning is best done when a learner can relate to the context of the problem. This activity asks students to predict and analyze data for a topic that they can easily relate to, understand and develop questions about.
 * What is being learned? What mathematics is the focus of the activity/technology? Is relational or instrumental understanding emphasized?**

The role of technology is access to information, simplifying tasks, and also representing knowledge and thinking. Students gain access to the data they need to compare in order to determine whose backpacks are too heavy from the Tinkerplots activity. Also, the sorting of the data and organization is simplified through the use of the technology. The students do not have to create new graphs or tables every time they want to compare the data and instead can use the technology to do that. Finally, the technology allows students to represent their thinking of how they would compare and organize the data. The advantages of the technology is ease of comparing all the data but also students lose out on the practice of creating their own graphs. The technology is unique because it is easy to look at many points of data in various ways in order to compare the information. The technology here supports such collaborative work because of its' dynamic ability to change based on the input and output values. Students are also able to view their data in a variety of ways.
 * What role does technology play? What instructional function(s) does the resource serve?**

This resource is centered on dynamic visual representation of data. This activity is intended to supplement existing curriculum of grades 6-8. It is an opportunity for students to practice reading and interpreting data in addition to the existing curriculum of data analysis. This lesson encourages students to work with partners or small groups to foster communication, create a variety of plots, and brainstorm alternate ideas. All the while, students are asked to share their ideas or plots with other students and explain how their plots were helpful in answering the questions. The technology here supports such collaborative work because of its' dynamic ability to change based on the input and output values. Students are also able to view their data in a variety of ways.
 * What kinds of representations of the mathematics are used?**

Geometer's Sketchpad
The developers of Geometer's Sketchpad intent was to provide a tool that can support and help students see that statements in Geometry need to be supported or refuted by evidence. The geometry standards stress the importance of analyzing characteristics of geometric shapes and forming relationships between shapes. It is very difficult to use a straightedge and protractor in a dynamic and interactive way. Geometer Sketchpads' developers have given students hands-on models and materials. They have provided students with a method for manipulating, making conjectures, and experiencing the "Beauty of Mathematics" when it comes to geometry. Students are able to use their conjectures to see if it still holds true in other examples. Geometer's Sketchpad would allow for students to work with one another to explore and understand their conjectures and discuss them with others in the class.
 * What is being learned? What mathematics is the focus of the activity/technology? Is relational or instrumental understanding emphasized?**

This technology provides students with a tool to construct meaning and an understanding of geometrical concepts. It provides quick and easy manipulations as well as visual representation of the angle changes and dimensions. This resource provides an instant calculation of the measurements and offers several pieces of information simultaneously. The dynamic software enables students to manipulate shapes, test conjectures, and understand properties of shapes in ways that would be impossible without such technology. The technology allows students to learn the characteristics of shapes by measuring angles, calculating slopes, and finding distances while manipulating shapes. Dynamic software also enables students to see patterns between different shapes and their intersections.
 * What role does technology play? What instructional function(s) does the resource serve?**

This resource would mostly serve as learning through exploration, though I think this resource could also be used to support direct instruction or presenting to students because of the visual and hands-on methods. This would be great for teachers who use SmartBoard's or similar technology in their classrooms.The dynamic nature of Geometer's Sketchpad allows for easy interface (i.e., you can select any of the possible products from the toolbar to create your geometrical shape and can do several calculations very quickly).
 * What kinds of representations of the mathematics are used?**

Brainingcamp: Pythagorean Theorem
Learn that the Pythagorean Theorem describes the relationship between the lengths of the sides of a right triangle. Discover visual and algebraic proofs. Apply the Pythagorean Formula to find missing side lengths. Derive the distance formula for points in the coordinate plane. This applet provides students with an opportunity to learn Pythagorean's Theorem with visual models and video tutorials with audio narratives. Next students explore interactively with examples that they can manipulate and test their conjectures. Students are then assessed with a questioning module to determine if students are understanding the concepts. Finally, this applet takes things a step further by applying and connecting Pythagorean's Theorem to real world context. This applets focus is on relational understanding with its' hands-on manipulative's and connecting ideas to real world applications. Learning takes place through exploration (i.e., provides context in which students can see new relationships; come to new understandings).
 * What is being learned? What mathematics is the focus of the activity/technology? Is relational or instrumental understanding emphasized?**

The technology provides the user with a precise, easy to follow instructional video, interactive lessons, assessment, and applications.Construction of triangles ,measurements, and the changes to the measurement of the triangles are performed instantly while the learner manipulates each side. The dynamic software enables students to manipulate the triangles, test conjectures, and understand properties of shapes in ways that would be impossible without such technology. This site also provides students who want to seek such knowledge with or without prior instruction on the topic.
 * What role does technology play? What instructional function(s) does the resource serve?**

The measurement of the sides of a triangle are represented symbolically through numbers, graphically on a coordinate plane, visually and spatially with its explanation of the theory, and dynamically through its interactive modules.
 * What kinds of representations of the mathematics are used?**

The Jasper Series
The Adventures of Jasper Woodbury consists of 12 videodisc-based adventures (plus video based analogs, extensions and teaching tips) that focus on mathematical problem finding and problem solving. Each adventure is designed from the perspective of the standards recommended by the National Council of Teachers of Mathematics (NCTM). In particular, each adventure provides multiple opportunities for problem solving, reasoning, communication and making connections to other areas such as science, social studies, literature and history (NCTM, 1989; 1991). This resource emphasizes relational understanding because it requires students to use their prior mathematical knowledge and skills to solve the problems.
 * What is being learned? What mathematics is the focus of the activity/technology? Is relational or instrumental understanding emphasized?**

Generally instructional videos consist of an instructor who is giving students answers or directions on how to solve problems. These series of video's provide students with just enough information needed to think critically about the different situations in each story. Students are then encouraged through communication with peers to solve problems, think critically, and use prior mathematical skills and concepts to solve each problem. This series is meant as a substitute or supplant to daily instruction. This unique series provides students with the necessary information to be successful in the problem solving as well as the ability to arrive at different answers.
 * What role does technology play? What instructional function(s) does the resource serve?**

These instructional videos connect mathematics in a concrete way with real world applications and communication.
 * What kinds of representations of the mathematics are used?**

** Smart Skies: Fly By Math **
FlyBy MathTM consists of five Air Traffic Control (ATC) Problems. Each examines a different distance-rate-time air traffic scenario that an air traffic controller might encounter. Each Air Traffic Control Problem features a Student Workbook containing an air traffic control experiment, paper-and-pencil calculations to support the experiment, and a student analysis of the experiment and calculations. The Workbook can be supplemented with optional pre- and post-tests. The FlyBy MathTM curriculum materials also include video clips to introduce students to the nation’s air traffic control system. The FlyBy MathTM curriculum materials have been developed by NASA’s Airspace Systems Program to engage students in Grades 5-9 in real-life applications of mathematics and science.
 * What is being learned? What mathematics is the focus of the activity/technology? Is relational or instrumental understanding emphasized?**

In addition to the FlyBy Math hands-on experiment and print worksheets, students can now run and solve each FlyBy Math problem electronically on the FlyBy Math simulator. The simulator uses the highest level of the 6 math approaches offered by FlyBy Math: graphing a system of linear equations. The FlyBy Math simulator is an online visualization tool that offers multiple linked dynamic representations to help students understand distance-rate-time relationships in the real-world context of air traffic control. The side-by-side layout enables students to observe and manipulate three linked views: the planes on their routes, the corresponding distance vs. time graph, and the equation of each line onthe graph. These experiments, lessons, and simulator supplement instruction and allow teachers to facilitate relational understanding while students explore these real life problems.
 * What role does technology play? What instructional function(s) does the resource serve?**

The instructional videos connect mathematics in a concrete way with real world applications. The simulators create a dynamic learning environment where students can manipulate and experiment with their calculations. This applet also provides graphical and visual/spatial representation of these distance-rate-time problems.
 * What kinds of representations of the mathematics are used?**