Promising iPod Touch and iPad Applications for Teaching and Learning Fractions By: Andrew Nicholas Undergraduate Research Project Virginia Tech Spring 2012 INTRODUCTION Technology should not replace teaching, but rather strengthen and support the overall lesson being taught in the mathematics classroom. However, effective technology use can only occur if the teacher understands how to use the technology to strengthen learning in the classroom. For mathematics teachers, technology should help students understand certain concepts, test their knowledge of a particular subject and be enjoyable and engaging. The use of iPod Touch and iPad apps has become prevalent among all subject areas so there are a lot of apps related to mathematics. Since there are so many apps for mathematics, how can teachers sift through them all and find ones that might strengthen learning for their students? The purpose of this paper is to share the development and application of a rubric that allows teachers to effectively and efficiently sift through iPod Touch/iPad apps specifically focused on Fractions concepts for grades 4-­‐8 and determine useful apps for supporting learning in the classroom. LITERATURE REVIEW To prepare my rubric, I turned to articles that discussed three main topics: the engagement and design of iPod Touch/iPad apps, incorporation between mathematics and iPod Touch/iPad apps, and standards for Fractions Concepts from grades 4-­‐8. First of all, I wanted to know what the required Fractions concepts were for grades 4-­‐8, as stated by the Virginia Standards of Learning (SOLs) and the Common Core State Standards (CCSS) (see Appendix A). I chose to focus more on the Virginia SOLs because I was presenting my research at the Virginia Council of Teachers of Mathematics (VCTM) Conference held in Roanoke, VA in March 2012. I also wanted to read about the CCSS and include its requirements in my research so that my findings could be applicable to teachers all over the country. Focusing on the Virginia SOLs, the standards stressed that students “will compare and order fractions and mixed numbers; represent equivalent fractions and identify the division statement that represents a fraction” (see Standard 4.2abc in Appendix A). The state wanted students to understand fractions as numbers; that they have a place on the number line in addition to other numbers. Also, the standards required that students be able to add, subtract, multiply and divide fractions, solve single and multiple problems involving fractions, investigate fractions as percents and ratios and be able to represent fractions from written and verbal information. The CCSS did not differ much from the Virginia SOLs, but one main difference was the CCSS stressed using previous knowledge to help build the Fractions Concepts up in the students’ minds. Using previous knowledge to help learn new concepts in mathematics allows the student to get more comfortable with the reviewed information as well as allow the student to develop his/her own understandings of what the new concept is. From reading about these Fractions Concepts, I developed a good basis for what I was to be expecting in an app that claimed to focus on certain Fractions concepts as described in the CCSS and the Virginia SOLs. “Perhaps one powerful reason for why almost a third of the students entering high schools in this country ‘drop out’ before completing their high school diploma (Gonzalez, 2010) is that education in many schools is presented in the same way as it was in the 19th and 20th centuries” (Olive 3). Technology needs to be incorporated into the mathematics classroom in this present day and age. Students use various types of technology all of the time outside of school, so, in his article, Olive was making it clear that the schools need to be incorporating technology as much as possible in this present time. Olive goes on to talk about the role of the learner and of the teacher in regard to using technology in the classroom. “As Steffe points out in his plenary paper, the researcher needs to be engaged with the students in a teaching-­‐learning situation in order to build second-­‐order models of students’ ways and means of operating when engaged in challenging mathematical tasks with the aid of technology” (Olive 11). If it is hard for the research/teacher to be engaged in helping his/her students with challenging mathematical tasks with the aid of technology, then either the teacher is not doing the right thing or the technology, specifically the iPod Touch/iPad app, may not necessarily be a good use of technology at the particular time. Olive makes the point that “research on how the specific interface design of a technology impacts its use is critical for understanding the ways in which humans interact with the technology” (Olive 11). Before the iPod Touch/iPad technology even gets used, teachers have to understand how it could be used to enhance students learning. This specific type of technology has to be research before it can even be used in the classroom. Olive stressed understanding how the teacher and the learner interact with each other with the aid of technology and he believes each type of technology should be evaluated on its design, content and accessibility. Shuler (2012) specifically dealt with Apple research on the education category for iPod Touch/iPad apps. Apple found that most of the educational apps were garnered towards young children (58%), with adults being the second most targeted audience at 40% (Shuler 3). However, only 18% of the educational apps are targeted for the middle school audience and only 10% for the high school aged students (Shuler 3). What does this mean? Finding useful and effective apps for Fractions concepts garnered towards grades 4-­‐8 will be hard to find. When considering the focus of all of the educational apps offered by Apple, 47% focus on early childhood learning, but the next biggest category is Mathematics at 13% (Shuler 13). Also, of the apps designed middle school students, 58% were intended to be used in schools (Shuler 13). From this article, there was some troubling information. Only 18% of all educational apps are designed for middle school students with 13% of those being focused on Mathematics. Finding apps that are free (this research only talked about free apps) and specifically focused on Fractions concepts will come few and far between. Getting these statistics helped me understand how hard it is to find effective iPod Touch/iPad apps to use in the classroom. “At first glance, the available apps for mathematics or for education seem to suggest that the most useful tool for iPods involves lots of practice of mathematical skills, especially those related to computation, with many of them focused on the primary years of schooling” (Kissane 937). Most apps that focus on mathematics have basic computations as their activity, while others go above and beyond and have students do mathematics. “Practice certainly has an important place in school mathematics, and a device that uses color, entertainment and novelty effects to engage students in practice at a range of levels may be a useful supplement to other experiences” (Kissane, 937). From this article, I learned that the majority of the apps just do basic mathematical calculations and are quite simple. There are very few applications that go beyond the scope of basic calculations. These articles gave me a firm basis for understanding the accessibility of apps teaching and/or supporting Fractions concepts, the engagement of apps, as well as truly knowing what the Fractions concepts were. For my project, having this firm basis helped me achieve my goal of answering my research question. METHODS To achieve my purpose of having teachers sift effectively and efficiently through iPod Touch/iPad apps focusing on Fractions concepts for grades 4-­‐8, I realized that I needed a tool to evaluate each iPod Touch app. The process of developing a rubric to evaluate apps to deem them “effective” or not came into being after reading the article “IQA Academic Rigor: Mathematics Rubric for Potential of the Task”, by Melissa D. Boston and Margaret S. Smith (2009). In this article, they talked about evaluating mathematical tasks given by teachers in the classroom. From their article, along with the Fractions concepts as stated by the VA SOL and CCSS (see Appendix A), I started developing criteria for my rubric. The first topic the apps were going to be evaluated on was Relevance to Mathematical Content (see Appendix B). The apps had to deal with Fractions concepts in some way. I came up with a level of Relevance scale from 0 to 4, each number representing a certain regard to how much the app actually focused on Fractions concepts. For example, the level of 4 was defined as “The app clearly focuses on Fractions standards as stated by VA SOLs and CCSS. “The app’s main focus is supporting fractions concepts explicitly”. The level of 0 was defined as “The app does not contain fractions at all”. This was my most important area of analysis because my whole goal was to find apps that talked about Fractions concepts for grades 4-­‐8. Boston and Smith use the Levels of Cognitive Demand as developed my Stein to rate the depth of each mathematical task given to the students. Since Stein clearly states the differing levels, I decided to make my next area of criterion be Level of Cognitive Demand (see Appendix B). Stein’s levels are Doing Mathematics, Procedures with Connections, Procedures without Connections and Memorization (Boston 152-­‐153). I used the same 0 to 4 point scale again, with Doing Mathematics receiving a 4 and proceeding down the levels until a score of “0” would be an app that requires no mathematical activity. Having apps that require students to think, make connections and develop understanding of Fractions concepts should definitely be rated very high as effective apps to use in the classroom. This second area of criterion was already stated for me so using the Levels of Cognitive Demand provided clear structure of evaluation. My third criterion, Engagement/Game Design, is the area I had to research the most. What does an engaging app look like? How to apps entice teachers and students alike so it actually aids teaching and learning? After reading numerous articles, I came up with another 0 to 4 point scale to rate the engagement of apps. Receiving a score of “4” in this area was defined as “The app has multiple windows with information, along with different steps to complete the game. Each frame in the game requires some work to move on to the next step. Also, the game is easy to follow and very clear in directions” (see Appendix B). A score of “2” for example would be defined as “The app has only a couple windows in the game with relatively simple tasks to complete. Also, the flow of the game may be even more confusing; with students struggling to know what is required of them”. Some games may be very clear in direction, but if they are only drilling the same thing over and over again, students are bound to get bored very quickly and not learn as effectively. This criterion was extremely important because using the iPod Touch app technology in the classroom effectively also depends on the design of the app itself. You could have an app that has a high level of cognitive demand and be strictly focused on Fractions concepts, but if it is the most boring or confusing app ever, students are not going to learn effectively and teachers are going to struggle to teach effectively. I developed a rubric with three areas of evaluation: Relevance to Mathematical Content, Level of Cognitive Demand and Engagement/Game Design (see Appendix B). With each level of each area carefully and clearly thought out, I felt moving forward that I developed an effective rubric that could rate apps so it could deem them effective and efficient in the classroom for teachers. When considering an app, I added up its score from each area to form a total score. An overall score of 10-­‐12 would be considered excellent apps for teachers to use. Apps with an overall score of 7-­‐9 would be considered decent to good apps to use in the classroom. Apps with an overall score of 0-­‐6 would not be considered good apps to use at all. A list of all of the apps I rated are available in Appendix C. ANALYSIS To show my analysis of the iPod Touch apps, I provide an example of my full evaluation of an app. The app I evaluated was Candy Factory by the Learning Transformation Research Group at Virginia Tech. In Candy Factory, you have to fulfill an order for a candy bar of a specified size. You have to break up a whole candy bar into pieces then iterate one piece a certain number of times in order to match the customer order. After playing the game and testing all aspects, I decided to give it a score of “4” in Relevance to Mathematical content. The app clearly focused on Fractions concepts 4.NF.1, 4.NF.3, 5.NF.3, 5.NF.5, 6.RP.1 and 7.RP.1 from the CCSS and Fractions concepts 4.2b, 6.2bcd, from the VA SOLs. This game brought together a lot of different standards, which is also another reason why I gave it a “4”. I gave Candy Factory a score of “4” for Level of Cognitive Demand because the game engaged students in exploring and understanding the nature of Fractions concepts using complex and unpredictable ways to solve the “issue” at hand, i.e. choosing the correct candy order to match the customer order. To play the game, the student has to identify patterns, make explicit connections and follow certain procedures. (Boston 152-­‐153) For the last criterion, Engagement/Game Design, I gave Candy Factory a score of “3” because the game had relatively simple screens and sometimes, the directions were not clearly stated in the game. When I observed people using the game, I had to show people where to go next in the game. If the directions were clearly stated, I would have given the app a score of “4”. Overall, Candy Factory received a score of 11, making it an excellent app to strengthen Fractions concepts among students in the classroom. I went through a similar process for each app I evaluated. First, I played the game and tried to explore as much as the game as possible to see all different scenarios. Once I had a good idea of what the game involved, then I evaluated the game based only on its Relevance to Mathematical Content. Then, I evaluated the game on its Level of Cognitive Demand and finally, evaluated it on its Engagement/Game Design. Once I had individual scores for each area of evaluation, I added the scores up to get the total score and determine whether the app was an excellent app (scores 10-­‐12), a good app (scores 7-­‐9) or an ineffective app (scores 0-­‐6). Having this system made the process of evaluating apps go smoothly. CONCLUSION All in all, I felt my research question was answered during my research. The purpose of this project was to develop and apply a tool that could be used by teachers to effectively and efficiently evaluate iPod Touch/iPad apps pertaining to Fractions concepts for grades 4-­‐8. From what I learned from the readings and through my evaluations of iPod Touch apps, I can say that the rubric I developed can be used to effectively evaluate apps so teachers know which ones to use in the classroom to aid in their teaching of Fractions concepts for grades 4-­‐8. My three areas of evaluation (Relevance to Mathematical Content, Level of Cognitive Demand and Engagement/Game Design) provided a good basis for determining which apps would be useful in the classroom. I can definitely say though that my purpose can be achieved if there is a rubric that can effectively evaluate apps, which I felt mine did. If I could do my research project all over again, I would change a couple things. First, I would have to change the scope of my Engagement/Game Design criterion. I focused my attention towards evaluating game design – not app design. I came across a couple apps that score very high in the first two areas of evaluation, but since there was no “game” aspect to it, I gave it a low score in that area, which, in turn, pulled down its overall score. For my rubric, I would add another criterion which would be focused on evaluating apps based on the cost efficiency of each. My research project only focused on free apps, but if I included apps that cost money, then I could develop an area of evaluation which rated apps on a cost-­‐effective, best deal ratio. Also, I would be interested in evaluating apps not just for Fractions Concepts, but for all different areas of Mathematics. All in all, I rated 19 free apps (see Appendix C). 5 were evaluated as excellent apps, 10 were evaluated as ok apps to use and 4 were evaluated as ineffective apps. In general, most apps are useful in the classroom, but there are few that are extremely effective in the classroom. I look forward to evaluating more apps, expanding my scope of evaluation and tweaking what I already have developed in order to make a rubric that is very useful for teachers to use in determining whether or not to use the iPod touch technology in the classroom. References Boston, Melissa D., and Margaret S. Smith. "IQA Academic Rigor: Mathematics Rubric for Potential of the Task." Journal for Research in Mathematics Education 40.2 (2009): 152-­‐153. Print. Kissane, Barry. MATHEMATICS EDUCATION AND THE iPOD TOUCH. Australia: AAMT & MERGA, 2011. Print. "Mathematics Standards of Learning." Virginia Department of Education. Virginia Department of Education, 2011. Web. 15 Feb. 2012. <http://www.doe.virginia.gov/testing/sol/standards_docs/ mathematics/index.shtml>. Olive, John. Research on Technology Tools and Applications in Mathematics Learning and Teaching. Athens, GA: n.p., 2010. Print. Shuler, Carly, Zachary Levine, and Jinny Ree. ILearn II An Analysis of the Education Category of Apple’s App Store. N.p.: Joan Ganz Cooney Center, 2012. Joan Ganz Cooney Center. Web. 15 Feb. 2012. <http://joanganzcooneycenter.org/upload_kits/ilearnii.pdf>. "The Standards: Mathematics." Common Core State Standards. Common Core State Standards Initiative , 2011. Web. 15 Feb. 2012. <http://www.corestandards.org/the-­‐standards/mathematics>. Vincent, Tony. "Education & Technology Quotes." slideshare. SlideShare Inc., 1 June 2009. Web. 15 Feb. 2012. <http://www.slideshare.net/tonyvincent/education-­‐technology-­‐quotes>. Appendix A: CCSS and VA SOLs Fractions Concepts Common Core State Standards Mathematics » Grade 4 » Number & Operations—Fractions¹ Extend understanding of fraction equivalence and ordering. • 4.NF.1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. • 4.NF.2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers. • 4.NF.3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b. o Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. o Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. o Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. o Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. • 4.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. o Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4). o Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) o Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? Understand decimal notation for fractions, and compare decimal fractions. • 4.NF.5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.2 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. •
•
4.NF.6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 4.NF.7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. Mathematics » Grade 5 » Number & Operations—Fractions Use equivalent fractions as a strategy to add and subtract fractions. • 5.NF.1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) • 5.NF.2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. Apply and extend previous understandings of multiplication and division to multiply and divide fractions. • 5.NF.3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-­‐pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? • 5.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.  Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)  Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. • 5.NF.5. Interpret multiplication as scaling (resizing), by:  Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.  Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers •
•
greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1. 5.NF.6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1  Interpret division of a unit fraction by a non-­‐zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.  Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.  Solve real world problems involving division of unit fractions by non-­‐zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-­‐cup servings are in 2 cups of raisins? Mathematics » Grade 6 » Ratios & Proportional Relationships Understand ratio concepts and use ratio reasoning to solve problems. • 6.RP.1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.” • 6.RP.2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”1 • 6.RP.3. Use ratio and rate reasoning to solve real-­‐world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.  Make tables of equivalent ratios relating quantities with whole-­‐number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.  Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?  Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. 
Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. Mathematics » Grade 7 » Ratios & Proportional Relationships Analyze proportional relationships and use them to solve real-­‐world and mathematical problems. • 7.RP.1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour. • 7.RP.2. Recognize and represent proportional relationships between quantities.  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.  Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.  Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. • 7.RP.3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. VA SOL Grade Four Number and Number Sense 4.2 The student will a) compare and order fractions and mixed numbers; b) represent equivalent fractions; and c) identify the division statement that represents a fraction. 4.3 The student will a) read, write, represent, and identify decimals expressed through thousandths; b) round decimals to the nearest whole number, tenth, and hundredth; c) compare and order decimals; and d) given a model, write the decimal and fraction equivalents. Computation and Estimation 4.5 The student will a) determine common multiples and factors, including least common multiple and greatest common factor; b) add and subtract fractions having like and unlike denominators that are limited to 2, 3, 4, 5, 6, 8, 10, and 12, and simplify the resulting fractions, using common multiples and factors; c) add and subtract with decimals; and d) solve single-­‐step and multistep practical problems involving addition and subtraction with fractions and with decimals. Grade 5 Number and Number Sense 5.2 The student will a) recognize and name fractions in their equivalent decimal form and vice versa; and b) compare and order fractions and decimals in a given set from least to greatest and greatest to least. Computation and Estimation 5.5 The student will a) find the sum, difference, product, and quotient of two numbers expressed as decimals through thousandths (divisors with only one nonzero digit); and b) create and solve single-­‐step and multistep practical problems involving decimals. 5.6 The student will solve single-­‐step and multistep practical problems involving addition and subtraction with fractions and mixed numbers and express answers in simplest form. Grade Six Number and Number Sense 6.1 The student will describe and compare data, using ratios, and will use appropriate notations, a
such as , a to b, and a:b. b
6.2 The student will a) investigate and describe fractions, decimals, and percents as ratios; b) identify a given fraction, decimal, or percent from a representation; c) demonstrate equivalent relationships among fractions, decimals, and percents; and d) compare and order fractions, decimals, and percents. 6.3 The student will a) identify and represent integers; b) order and compare integers; and c) identify and describe absolute value of integers. 6.4 The student will demonstrate multiple representations of multiplication and division of fractions. Computation and Estimation 6.6 The student will a) multiply and divide fractions and mixed numbers; and b) estimate solutions and then solve single-­‐step and multistep practical problems involving addition, subtraction, multiplication, and division of fractions. Grade 7 Number and Number Sense 7.1 The student will a) investigate and describe the concept of negative exponents for powers of ten; b) determine scientific notation for numbers greater than zero; c) compare and order fractions, decimals, percents, and numbers written in scientific notation; d) determine square roots; and e) identify and describe absolute value for rational numbers. Grade 8 Number and Number Sense 8.1 The student will a) simplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers; and b) compare and order decimals, fractions, percents, and numbers written in scientific notation. Computation and Estimation 8.3 The student will a) solve practical problems involving rational numbers, percents, ratios, and proportions; and b) determine the percent increase or decrease for a given situation. Appendix B: Rubric OVERVIEW: The purpose of this rubric is to allow teachers to efficiently and effectively evaluate iOS – based applications pertaining to fractions concepts. Specifically focusing on the Virginia SOLs for grades 4-­‐8, along with the Common Core State Standards for grades 4-­‐8, this rubric will set guidelines for necessary characteristics of effective educational games. EVALUATION: Each app will be evaluated on a 0 to 4 scale in 3 different categories as follows: 1. Relevance to Mathematical Content – Evaluation of alignment with subject area standards as specified by attached SOL and CCSS Grade 4-­‐8 Standards. • 4 – The app clearly focuses on fractions standards as stated by SOLs and CCSS. The app’s main focus is supporting fractions concepts explicitly. • 3 – The app addresses the same standards as above, but not as the main focus. You still have to do a lot of work with Fractions concepts, but you still achieve a different goal in the game. • 2 – The app addresses fractions concepts in a minimal way. Similar to 3, but the difference is a main focus of the game is not related to mathematics at all. • 1 – The app does not support fractions concepts at all. There are fractions in the game, but they are addressed superficially. • 0 – The app does not contain fractions at all. 2. Level of Cognitive Demand – Each app will be evaluated based on what levels of processing it requires. • 4 – The game has the potential to engage students in exploring and understanding the nature of mathematical concepts, procedures, and/or relationships such as: doing mathematics – using complex and nonalgorithmic (i.e., there is not a predictable, well-­‐
rehearsed approach or pathway explicitly suggested by the game, game instructions, or a worked-­‐out example), or applying procedures with connections – applying a broad general procedure that remains closely connected to mathematical concepts. The game may require students to develop explanations, identify patterns, make conjectures, make explicit connections among representations or follow a prescribed procedure. (Boston 152) • 3 – The game has the potential to engage students in complex thinking or in creating for mathematical concepts, procedures and/or relationships. However, the game does not receive a 4 because it does not explicitly prompt or evidence of students’ reasoning and understanding, students may be asked to engage in doing mathematics or procedures with connections, but the underlying mathematics in the task is not appropriate for the specific age group, students may need to identify patterns but are not pressed for generalizations, students may be asked to use multiple strategies or representations, but that task does not explicitly prompt students to develop connections between them and students may be asked to make conjectures but are not asked to provide mathematical evidence or explanations to support conclusions. (Boston 152) • 2 – This game does not require students to make connections to concepts or meaning underlying the procedures being used. The focus of the task appears to be on producing correct answers rather than developing mathematical understanding, or the task does •
•
not require student to engage in cognitively challenging work; the task is easy to solve. (Boston 153) 1 – The potential of this game is limited to engaging students in memorizing or reproducing facts, rules, formulae, or definitions. This game does not require students to make connections to the concepts or meaning that underlie the facts, rules, formulae, or definitions being memorized or reproduced. (Boston 153) 0 – The game/task requires no mathematical activity 3. Engagement/Game Design – Each app will be evaluated according to its complexity. This may be similar to level of difficulty, but this focuses on the actual game design instead of the content. • 4 – The app has multiple windows with information, along with different steps to complete the game. Each frame in the game requires some work to move on to the next step. Also, the game is easy to follow and very clear in directions. • 3 – The app has different challenges and windows as in level 4, but some windows may be confusing and the app is not as challenging. • 2 – The app has only a couple windows in the game with relatively simple tasks to complete. Also, the flow of the game may be even more confusing; with students struggling to know what is required of them. • 1 – The app has maybe one or two windows to the game and it is so simple that students are likely to lose interest quickly. • 0 – The game is not appropriate for the targeted student population. CONCLUSION: These 3 evaluation areas: relevance to mathematical content, level of cognitive demand and engagement/game design are vital to have in a great app for students to learn fractions in a positive and constructive way. If any of these areas’ scores is extremely low, then the app could be ineffective in supporting students’ learning. After evaluating each app with this rubric, apps with an overall score of 10-­‐12 will be considered excellent apps for teachers to use. Apps with an overall score of 7-­‐9 will be considered decent to good apps to use, but not as the only app to use in the classroom. Apps with an overall score of 0-­‐6 will not be considered good apps to use at all. When looking at an excellent or good app to use in the classroom, also focus on the app’s scoring in each category of evaluation. You may have an excellent app with a rating of 10, but its level of cognitive demand could be a 2. You may have a good app with an overall score of 7 but its level of relevancy could be a 1. Paying attention to the individual scores as well as the overall score will provide some insight on whether to use certain apps individually in the classroom or combine apps with each other to provide a great spread of the categories talked about in the rubric. These apps are meant to enhance teaching, not replace it. Using an app effectively in the classroom can help students learn material in ways they can understand. To show the necessity of using technology properly in the classroom , Nancy Kassebaum, a U.S. Senator says “There can be infinite uses of the computer and of new age technology, but if the teachers themselves are not able to bring it into the classroom and make it work, then it fails” (Vincent 2009)