**How is the user of your manipulative supposed to learn from working with your design?**

Studies have shown that “students who use concrete materials develop more precise and more comprehensive mental representations, often show more motivation and on-task behavior, understand mathematical ideas, and better apply these ideas to life situations,” (Hauser).

By using this manipulative that I have designed, I am hoping that the user will gain a better understanding of fractions. Fractions are often taught using 2-dimensional figures, graphics or manipulatives. Using 3 dimensions, students can better understand fractions through increased spatial awareness while introducing area and volume concepts at an earlier age if desired, as well. Using the blocks will give the student the chance to explore how a whole can be divided into equal parts. Additionally, the blocks can also be used to explore equivalent fractions or missing parts of the whole.

**How does the design of your technology reflect your understanding of what mathematics is and of how learning happens? **

In order to succeed in mathematics, I believe that students must develop an understanding of abstract concepts first. Additionally, learning the concepts of fractions specifically can be one of the most difficult skills to master for elementary level students. With so many different ways to expose students to manipulatives and enhance their fraction learning experience, it is important to examine how effective these teaching tools can be first.

These concrete materials are meant to assist children at all levels of education including understanding processes, communicating their mathematical thinking, and extending their ideas to higher order thinking levels (Balka, 1993). In addition, using manipulatives enables students to make connections with other mathematical topics, gain insight to other academic subject areas, and in their personal interests and experiences (Lee & Chen, 2010).

Therefore, just like I believe that fractions can be simple to understand, this manipulative is intended to be simple to understand. Also, just like math is meant to make our lives easier, so are the use of these manipulatives. Sometimes all it takes is to see something from a different perspective for it to make sense. Every child learns differently and children understand differently, so by giving the child a chance to make sense of something on their own terms, makes a huge difference.

**What would it mean for your project to be successful in terms of learning?**

If I were able to get a student to verbalize mathematical thinking while discussing mathematical ideas and concepts, and gain confidence in their ability to find solutions to mathematical problems using methods that they came up with themselves without relying on direction from myself or the teacher, then I would consider it to be successful in terms of learning.

With this project, my intention is to give the child a way to learn about fractions, while making the learning process more fun and engaging. I am also attempting to demonstrate a significant amount of student growth when being assessed. Furthermore, after using my design, I would like the students to be more willing to share ideas and explore concepts when it comes to fractions, as well as ask and answer higher order thinking questions.

Finally, in order for this manipulative to be considered successful in terms of learning, I would like the use of it to allow students to communicate their thoughts and ideas about fractions and what they have learned to other classmates, who may still be struggling, more easily.

**What would it mean if it were to fail to work?**

If this manipulative fails to work, it would mean that it was not an efficient enough tool for use. The goal is to help children understand the idea of fractions while building a strong understanding of mathematical concepts. If this manipulative cannot efficiently do that, then it has failed. Additionally, the idea of a manipulative is to correctly portray the actual math lesson or concept intended, while instilling a sense of understanding to the given topics. This means that if the manipulative is not effective, then it will not help the student understand the mathematical concept, and instead it may even confuse them further. Not to mention, we would like students to be able to use the manipulative in order to further their understanding so that they are able to explain it more clearly to other students who are still struggling, not lead them astray.

Finally, if after using this design, students still can’t enhance their achievements in mathematics, specifically when it comes to lessons regarding fractions, then it has failed to work.

**What kind of testing could you imagine carrying out for your design?**

I would use my design for a number of different tests. First, I would test out this manipulative with teachers. I would see what they think of it, and ask for any feedback on what is it about the product they initially like and don’t like. I would then ask if they would try it out either in their classroom, or with students they may tutor independently or after-school. Then I would collect more feedback from them based on how it helped their students or didn’t, and if they would consider continuing to use them. This information would also be further used in case studies I would complete regarding the effectiveness of this design.

Then I would like to test it on children, out of school first, preferably with a variety of different age and grade levels. I would like to see what the children’s initial thoughts about the product are, un-relative to math. (What they think it is, do they like the colors, aesthetics of it, what they think it is used for?) After that, I would introduce it to those children as a manipulative used to understand mathematical concepts, and explain how it can be used. I would also use this information for my case studies that I would be completing regarding this specific design.

Finally, depending on the information I gather from the first two test subjects, I would like to carry out some tests using this product with parents. Often these are the people who work with their children the most, and it is important to see if this is something that they find useful or not. It would be interesting to get the thoughts and opinions about the manipulative, from adults who potentially have little understanding themselves, of the concepts of mathematical ideas. Was it something they were able to easily understand? Was it something that they were easily able to explain to their children when helping with things like homework and studying? Again, this would be included in my case study that I would be doing for this design.

__REFERENCES__

Balka, D. S. (1993). Making the connections in mathematics via manipulatives. Contemporary Education, 65, 19-23.

Hauser , Jane. Concrete-representational-abstract instructional approach. Retrieved April 9, 2009, from the Access Center: Improving Outcomes for all Students K-8. Web site: http://www.k8accesscenter.org/training_resources/CRA_Instructional_Approach.asp

Lee, C., & Chen, M. (2010). Taiwanese junior high school students’ mathematics attitudes and Perceptions towards virtual manipulatives. British Journal of Educational Technology, 41(2), E17-E21. doi:10.1111/j.1467-8535.2008.00877.x