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Dynamic Software in Geometry
Transcript of Dynamic Software in Geometry
The Standards mandate that eight principles of mathematical practice be taught:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning. The purpose of this study is to demonstrate the effects of the use of dynamic software on students’ achievement on traditional assessments in Geometry classrooms. The participants in this study includes two high school Geometry honors level classes and two high school Geometry academic level classes. One of each level would be taught using traditional methods and the other two classes would be taught with the use of dynamic software programs such as Geometer’s Sketchpad. The two classes that would use the software would be introduced to the program and then complete self-guided constructions and make inferences based on their findings. The other two classes would use traditional hand-tools to make these same constructions and make their own inferences based on their findings. Both classes would be assessed using traditional methods and the results would be compared. Abstract Research Two themes emerged from the review of the literature related to the role of dynamic software in students’ understandings of geometry:
1. The idea of varying spatial ability.
2. The ways in which computer environments mediate students’ understandings of geometry.
The former is useful for describing how students reason about geometry, whereas the latter focuses more directly on the technology tool. Spatial Ability -Changing, rotating, bending, and reversing an object presented for stimulating the mind
-Mental processes being used in perceiving, storing, recalling, creating, arranging, and making related spatial images.
-Understanding relations visually, making changes on shapes, rearrangement and interpreting them. The Development of Spatial Sense Cannot be taught
Developed over a period of time
Engage in spatial activities prior to manipulating images
Spatial orientation vs Spatial insight
van Hiele van Hiele levels of Geometric thought describes how children learn to reason in geometry
students need to build an extensive of the systems of geometric ideas
developed by familiarity through examples/counterexamples, relationships between properties, and how properties are ordered Level 0: Visualization At this level, the focus of a child's thinking is on individual shapes, which the child is learning to classify by judging their holistic appearance. Level 1: Analysis At this level, the shapes become bearers of their properties. The objects of thought are classes of shapes, which the child has learned to analyze as having properties. Typical Student Response: "That is a circle," usually without further description. Children begin to notice many properties of shapes, but do not see the relationships between the properties; therefore they cannot reduce the list of properties to a concise definition with necessary and sufficient conditions. They usually reason inductively from several examples, but cannot yet reason deductively because they do not understand how the properties of shapes are related. Typical Student Response: Level 2: Abstraction At this level, properties are ordered. The objects of thought are geometric properties, which the student has learned to connect deductively. The student understands that properties are related and one set of properties may imply another property. Students can reason with simple arguments about geometric figures. Typical Student Response: "Isosceles triangles are symmetric, so their base angles must be equal." Level 3: Deduction Students at this level understand the meaning of deduction.
The object of thought is deductive reasoning, which the student learns to combine to form a system of formal proofs.
Learners can construct geometric proofs at a secondary school level and understand their meaning.
They understand the role of undefined terms, definitions, axioms and theorems in Euclidean geometry.
However, students at this level believe that axioms and definitions are fixed, rather than arbitrary, so they cannot yet conceive of non-Euclidean geometry.
Geometric ideas are still understood as objects in the Euclidean plane. Level 4: Rigor At this level, geometry is understood at the level of a mathematician.
Students understand that definitions are arbitrary and need not actually refer to any concrete realization.
The object of thought is deductive geometric systems, for which the learner compares axiomatic systems.
Learners can study non-Euclidean geometries with understanding.
People can understand the discipline of geometry and how it differs philosophically from non-mathematical studies. Sang Sook Choi-Koh 1986
Investigated the geometric learning of a secondary school student based on van Hiele model.
Instructed through the use of dynamic computer sofware
Collected data twice a week through assessment questions Results: Summary: Period 1: Able to analyze figures in terms of their properties and relationships
Through active discussion and use of GSP, the student was able to orient himself in geometric thought. Period 2: Independent in identifying properties and developing informal arguments
Used active visualization to take advantage of constructing an object to interrelate previously discovered properties Period 3: Able to demonstrate the development of a logical, deductive system of geometric thought.
Developed reasoning for formal deduction based on previously discovered concepts. Overall: GSP was an effective tool for this student to explore the implicatory properties of relationships.
A well-designed computer environment is useful enough to help students build connections between the intuitive and analytical aspects of mathematical objects and operations.
The software allows a wide range of open-ended investigations to promote the ability for conjecture and mathematical thinking. Geometer's Sketchpad Additional Research Roberts, D.L. & Stephens, L.J. (1999) Studied 3 Geometry classes in an urban high school
Looked at results from a motivational and learning perspective
Research Question: Does the use of dynamic software increase the interest and enjoyment of the students?
Results: The study showed that the use of software improved student interest and participation in Geometry Charles Funkhouser (2002) Similar study
Researcher was interested in whether the use of dynamic software positively impacted students' attitudes towards mathematics.
Results: Showed no statistically significant difference between the treatment and control groups but statistically significant results showed that students taught with geometry software performed at a higher level of achievement on traditional assessments than those in the control group Amy Hull (2004) Research Questions: What is the effect of the use of dynamic geometry software on students' scores on traditional geometry assessments?
What is the effect of the use of dynamic geometry software on students' attitudes towards geometry? At the beginning of the study students completed an attitude questionnaire. The questionnaire was designed to give me an idea of student attitudes towards geometry prior to the start of the study.
During the course of the study classes were taken to the computer lab twice a week. While in the lab students completed activities designed to help them discover important geometric properties that relate to circles. Students were asked to answer various questions that required them to critically think about the properties. Students then made conjectures based on the investigations and answers to the questions.
At the conclusion of the study students completed the same student attitude questionnaire that was completed at the beginning of the study. Results: The study showed that the results the assessment were comparable to the previous year.
Students felt that this type of exploration was more convincing than formal proof and enjoyed lessons based in the computer lab. (Dixon, 1997; Gerretson, 2004; Myers, 2009) Results obtained from the statistical analyses of these researches suggested that students experiencing the dynamic geometry instructional environment significantly outperformed students experiencing a traditional environment on content measures of the concepts and skills taught during the experiments. Hannafin, Burruss, & Little (2001) The results showed that the teacher had difficulty relinquishing control of the learning environment, while students enjoyed their new freedom, worked hard, and expressed greater interest in the mathematics content. Jiang, Z. (2002) Article describes how the use of the dynamic geometry software has helped teachers develop their abilities in three aspects:
1. challenging problem solving
2. mathematical modeling
3. constructing student-centered teaching projects. A course at NYU was given to mathematics teachers on how to appropriately implement dynamic geometry software into the classroom Results: Teachers have benefited from exploring mathematics and mathematics teaching with technology and especially GSP.
The examples given point out that for some of the challenging problems that are presented to the students, it is almost impossible or very hard to manually make correct drawings.
The use of dynamic geometry software seems to be critical, or at least very desirable.
The use of the software can simulate students' insight of problem solving and provide an easy and convincing way of verifying the solution.
Students can construct accurate visual representations to model real world situations very efficiently by using transformations in dynamic geometry software.
Good projects that take advantage of dynamic geometry software can also effictively enhance students' mathematics learning. Myers, Ron (2009) Examined the effects of the use of technology on students' mathematics achievement on the Florida Comprehensive Assessment Test
Eleven schools participated in a pilot program on the use of Geometer's Skethcpad. Three of these schools were randomly selected for this study.
In each school, the GSP class and a traditional geometry class taught by the same teacher were the study participants. Results: The findings of the study revealted a significant difference in the FCAT mathematics scores of students who were taught using GSP compared to those who used the traditional method. Features The Geometer's Sketchpad is the world's leading software for teaching mathematics. GSP gives students at all levels-from third grade through college-a tangible, visual way to learn mathematics that increases their engagement, understanding, and achievement. http://www.youtube.com/embed/EGXf4hZtVuo?showinfo=0&modestbranding=1&autoplay=1&autohide=1&rel=0&loop=1&playlist=EGXf4hZtVuo Elementary school:
1. Dynamic models of fractions
2. Number lines
3. Geometric patterns Middle School:
1. Ratios and proportions
2. Rate of change
3. Functional relationships
4. Graphical representations High School:
1. Construct and transform geometric shapes.
2. Linear, quadratic, and trigonmetric functions
3. Deeper understanding Data Setting The setting for this study is going to be at Mark T. Sheehan High School in Walliingford, CT.
1. Honors Level - 2 classes
2. Academic level - 2 classes Purpose The purpose of this study is to describe the effects of the use of dynamic geometry software on students' achievement on traditional assessments. In doing this, I hope to determine how to best use the software program in cultivating the curriculum to align with the CCS. The experimental group will be one Geometry H and one Geometry A class that uses Geometer's Sketchpad during instruction. The control group wll be the pair of classes that use traditional teaching strategies. Research Questions 1. What is the effect of the use of dynamic geometry software on students' scores on tradititonal geometry assessments?
2. What is the effect of the use of dynamic geometry software on students' attitudes towards geometry?
3. Is there an interaction between the use of dynamic geometry software and the level of the students? Questions?