Introduction: Standards, Goals and Objectives of the Project
Virtual manipulatives/online software (VM/OS) was introduced to me in my college math classes. I enjoyed using them because they allowed me to have access to manipulatives that I otherwise would not have had. They exposed me to the different types of hands on manipulatives that students will use in the classroom, whether as concrete objects or virtually. I also analyzed online software such as applets. Since then I have not used any online software but I have used virtual manipulatives in my classroom, both as a 5th grade student teacher and a mathematics paraprofessional. As a student teacher, I used a virtual Geoboard on the SMARTBoard to model creating triangles then the students used actual Geoboards in partner work to work on solving problems. The virtual Geoboard allowed me to model how to use the Geoboard, model how I wanted the students to use it, and have students practice using it in front of the whole class, with much higher viability than if I was just using a regular sized Geoboard. As a paraprofessional, I have had students use virtual manipulatives mainly for fraction work and for balancing equations. There are many wonderful fraction kits online as well as scales which balance shapes, until the student figures out what an individual shape equals. Most of the virtual manipulatives I have used came from the National Library of Virtual Manipulatives, as they have a large selection of manipulatives that are elementary appropriate. I have also used Google SketchUp. It was initially introduced to me by a math teacher who was using it in her STEM Club. I downloaded and love playing around with it, although I have yet to use for anything instructional or mathematical as suggested by Livingston and Fearon (2012).
The actual term manipulative is not used in the NCTM standards or the CCSS for math. VM/OS however have large potential in the role of teaching of elementary mathematics. One of the guiding principles under Algebra is “Use mathematical models to represent and understand quantitative relationships” with correlated standards for Pre-K-12. Some of these include the Pre-K -2 standard “model situations that involve the addition and subtraction of whole numbers, using objects, pictures, and symbols.” There is also the Pre-K to 2 Numbers and Operations standard of “connect number words and numerals to the quantities they represent, using various physical models and representations” and “use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals.” Using modeling is also mentioned in terms of geometric models where students are expected to “Use visualization, spatial reasoning, and geometric modeling to solve problems” (NCTM, 2000). Within the Common Core, as a part of being able to reason abstractly and quantitatively, are expected to be able “to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents” (CCSS, 2010, p. 6). Students are also expected to manipulate units and fractions.
Just as in Kaplan and Alon (2013), where the teacher sought to build the student’s knowledge of relationships and knowledge of the numbers through use of the manipulatives, use of fraction manipulatives in this activity rather than just teaching the steps is set up to lead to a higher conceptual understanding of the process by the students rather than just having knowledge of the steps, which are hard to retain (p. 387). As Johnson, et al. (2012) stated picking appropriate tools is important part of the process. It is important to pick tools that “expose a deeper level of student understanding and support a higher level of questioning than other tools that addressed the same content” (p. 204). There are also other aspects of virtual manipulatives that are important to consider. Tools need to be able to have multiple abilities and be able to be adapted the nuances of students’ abilities. Johnson mentions how a teacher
As explained by Suh (2005) “manipulatives and other tools are not sufficient on their own…teachers should guide students in building understanding with manipulatives and other tools with meaningful representation of mathematical concepts” (p.15). Suh also raises the objection that manipulatives might mask a student’s understanding of mathematical concepts as it makes the math too easy for the student. However, with the proper teacher instruction, manipulatives can be used to scaffold student learning and check for student learning during and after manipulative use.
The purpose of this project is to explore the use of fraction manipulatives in teaching the concept of multiplying fractions by fractions at a fifth grade level. More specifically, the project has the following objectives:
1. Students will visually be able to see the effect of multiplying fractions to gain an understanding of the reasoning behind the answer and not just the process.
2. Students will apply the process of multiplying fractions, building on the prior knowledge of adding fractions and of multiplying fractions.
Overview of the VM/OS Project
This project was initially implemented in a one-on-one intervention setting with an 8th grade student at the beginning of the school year. However, this project was adapted to be used with a small groups of fifth grade students. The project is focused around CCSS standards 5.NF.4b: “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” (CCSS, 2010, p. 36). Students working to meet this standard will complete Worksheet 2. Students who have not ready for this standard will work towards mastering CCSS standards 5.NF.4 “Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction” as well as the subsets of this standard 5.NF.4a: “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.)” (CCSS, 2010, p. 36). These students will complete Worksheet 1. Students who have already mastered 5.NF.4b will work towards CCSS standard 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” (CCSS, 2010, p.36). These students will complete Worksheet 3.
This project, through the use of virtual manipulatives, will help students see what multiplication of one fraction by another fraction actually looks like – what effect does it have on the answer? How are the numbers and the sections related to each other? What do you know about multiplication and arrays that translates to multiplying fractions? The use of virtual manipulatives will lead to a conceptual understanding of multiplying fractions which will ideally help them retain their understanding of how to multiply fractions.
As mentioned by Johnson, et. al (2012), a teacher selected one tool over another because it allowed the teacher to manually enter the problems (p. 204). Likewise, the NVLM manipulative was selected because it allowed me to be able to have work shown, or not, and to switch from proper fractions to improper fractions which led to it being able to be used for a wider range of standards. It also allowed the students to be able to input their own fractions up to a certain point, as the denominators available were limited. Also mentioned by Johnston are the drawbacks to manipulatives, manly being “a lack of options for teacher input (p. 205). This came up when selecting manipulatives for the students to use in my activities. The NLVM manipulative would not work with Worksheet 1 because Worksheet 1 included whole numbers. The NLVM manipulative did not allow for the use of whole numbers in a sense where students of a lower abilities level would understand (although technically you could select the “improper fraction” option, then have students choose 2/1 as their fraction in place of 2, I felt developmentally this did not add to students understandings). Therefore, I choose the app from ABCYa for Worksheet 1 as it allowed students to work with virtual fraction strips. A computer could be used for accessing the NLVM, but a tablet device that used apps would be needed for the ABCYa app. In the model on the worksheet, an iPad was used, but I am assuming the app would also be compatible with other tablet devices.
As Suh (2005) points out “deliberate attention must be paid to help students transfer what they know in the context of the manipulatives to other representations, including symbols, numbers, and graphs” (p. 16). Therefore the one on one student interviews after finishing each worksheet, ensure that students are verbally able to explain the steps in solving a problem as well as use this new knowledge to solve a problem without using the manipulative.
The students will be given time to explore their virtual manipulatives (ABCya Fractions-Decimals-Percents virtual manipulative app for Worksheet 1; NLVM Rectangle Multiplication Manipulative for Worksheet 2 and 3) building their understanding of its capabilities so they can use it fluently when it comes to solving the problems, using other students in their group as problem solvers and assistants when they have a question.
The students will replicate the model provided in problem one, to build their knowledge of multiplying fractions, and also to gain an understanding of teacher expectations, in terms of how to set up the math problems and gaining familiarity with using the app for math purposes.
The students will then complete problem two on their own, basing it on the model and finding an answer. Students will be expected to save a screenshot of their work once they finish the problem, so the teacher can more thoroughly check student process at a later time.
Students will again follow the model provided by the teacher making the transition from switching multiplication to addition to actually multiplying the fractions in Worksheet 1; making the transition from seeing how the problem is solved to setting it up on their own for Worksheet 2; and making the transition from multiplying fraction to multiplying mixed numbers.
Students will then complete two problems on their own using the app.
Once the students finish their work, students will meet with the teacher. Students who completed Worksheet 1 will explain to the teacher how to rewrite 5 x 3/7 and explain how to find an answer. Students who completed Worksheet 2 will explain the process of multiplying two fractions (2/3 x 4/5) and show how to find an answer. Students, who have completed Worksheet 3, will discuss with the teacher the steps in multiplying mixed numbers, comparing it to multiplying fractions as well as demonstrating how to solve a problem.
Students will be assessed using the following checklist. The checklist will serve as a guide for assessing whether students are mastering the topic and whether they are ready to move on to a new topic.
- Completed all math problems on the worksheet correctly.
- Provided answers for all written questions on their worksheet.
- Screenshots demonstrated a mastery of the content
- Was able to verbally explain how to solve a problem in response to a prompt.
- Was able to demonstrate solving a sample problem without using the manipulatives.
I have taught this topic before in intervention settings. When I taught the topic I used a mix of worksheets, instructional videos and the NLVM website. These activities are based on the problems I used and Worksheet2 is most like what I have taught in terms of content and technology use. I however, have never used them in a fifth grade classroom. When working with fifth graders their mathematical knowledge might be lower than with the students receiving math interventions. Because of this, I am worried that I made too many broad jumps and assumptions about their knowledge in the worksheets. For example, if students cannot know how convert mixed numbers to improper fractions, reading the instructions on the worksheet might not be enough for them to understand the process. I think the set-up of the project, choosing three standards close to each other within the same grade, as well as using different manipulative types for specific purposes, is beneficial to students as in a given classroom students tend to be slightly staggered within their knowledge of grade level content.
ABCYa. Virtual manipulatives: fractions-decimals-percents app. Available from: https://itunes.apple.com/us/app/virtual-manipulatives!/id471341079?mt=8
Common Core State Standards Initiative. (2010). Common core state standards for mathematics. Available from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
Johnson, P. E., Campet, M., Gaber, K., & Zuidema, E. (2012). Virtual manipulatives to assess understanding. Teaching Children Mathematics, 19(3), 202-206.
Kaplan, R. G., & Alon, S. (2013). Using technology to teach equivalence. Teaching Children Mathematics, 19(6), 382-389.
Livingstone, J., & Fleron, J. F. (2012). Exploring three-dimensional worlds using Google SketchUp. Mathematics Teacher, 105(6); 469-473.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. Available from http://standards.nctm.org/.
National Library of Virtual Manipulatives. Fractions – Rectangle Multiplication. Retrieved from http://nlvm.usu.edu/en/nav/frames_asid_194_g_3_t_1.html?from=search.html
Suh, J. M. (2005). Third graders’ mathematics achievement and representation preference using virtual and physical manipulatives for adding fractions and balancing equations. (Unpublished doctoral dissertation). George Mason University, Fairfax, VA.