# Melissa Connor's Portfolio Page for Spring 2015 Mini-Course

## Contents

**My Topic/Purpose**

This course, *Using Visualizations to Teach Mathematics with Understanding*, will focus on how teachers can use and create visualizations to teach mathematics with understanding. Creating meaningful visualizations can enhance learning in the classroom by helping students conceptualize the big ideas of our lessons. When we use visualizations in the classroom, our students begin to develop their own visualization skills to help them solve problems. These skills can help our students become active and effective learners. In this course, participants will answer and reflect on the following questions:

- What does it mean to teach/learn with understanding?
- What are the advantages of teaching/learning with understanding?
- What are the differences between reform teaching methods and traditional teaching methods?
- What can teachers do to promote learning with understanding?
- What are visualizations and why are them important in mathematics curricula?
- What do effective and ineffective visualizations look like?
- How do you determine if a visualization is an accurate representation for a given problem?
- How do you create meaningful visualizations to supplement mathematics instruction?
- What are some tools and strategies you can use to create visualizations to teach mathematics with understanding in the classroom?
- How can we transfer these skills to our students?

By the end of this course, participants will develop their skills to recognize and create quality visualizations to teach mathematics with understanding.

**Learning Outcomes**

This course will help teachers develop effective visualization skills and provide them with tools and strategies to teach mathematics with understanding. By the end of the course, learners will:

- Conceptualize the idea of teaching and learning with understanding.
- Identify what visualizations are.
- Identify characteristics of meaningful and effective visualizations.
- Develop skills to create visualizations.
- Explore tools and strategies for using visualizations in the classroom and how to transfer these skills to students for his/her own use.

**Needs Assessment/Learner Analysis**

**Problem**

As quickly as the years pass, expectations for American mathematics students become higher and the curriculum becomes more rigorous. The National Common Core Standards promote high-level mathematical thinking and reasoning. Such high-level mathematics curricula “[value] students finding patterns, making connections, communicating mathematically, and engaging in real-life, contextualized, and open-ended problem solving […] with reduced emphasis on routine arithmetic computation” (Goldin, 2002). These skills can lead to learning with understanding and visualizations can be used to facilitate and organize ideas and learning goals.

Carrying this idea forward, teachers need to understand how students learn and the “conceptual models, technologies, principles, and reasoning processes” that exist to implement meaningful learning experience (Goldin, 2002). Visualizations help students to think about mathematical ideas and develop their own methods for solving problems.

**What is to be Learned**

Participants will learn what it means to teach and learn with understanding and how visualizations can supplement mathematics instruction to reach this goal in the classroom. Participants will determine the meaning of visualizations and why they are important in mathematics instruction. Additionally, participants will examine research and develop strategies for creating meaningful visualizations. Finally, participants will explore methods and instructional technology to not only provide students with meaningful visualizations, but also help students develop skills to create their own visualizations.

**The Learners/Learner Analysis**

Participants will be a combination of in-service/pre-served teachers, school administrators, and information technology specialists in the K-12 mathematics education environment. This mini-course will have an emphasis on secondary education. This mini-course will serve as a professional development resource and/or opportunity for participants. Participants will have some familiarity with using visualizations in mathematics but will be given the opportunities explore, share, and develop new ideas to reform and improve education in the 21st century to encourage learning with understanding.

**Instructional Context**

Participants will access this mini-course online through “The Knowledge Network for Innovations in Learning and Teaching” (KNILT), a wiki-based project directed by Professor Jianwei Zhang of the University at Albany in New York. Therefore, a computer with internet access will be mandatory.

**Exploring the Instructional Problem and Solution**

Considering my experience, I know that all students learn differently. Research shows students that struggle with computation skills “sometimes show strong visual, spatial, or logical reasoning ability in less routine mathematics” (Goldin, 2002). Edens and Potter state, “Visual-spatial representations may function as mental imagery […which can] assist students to learn to ‘unpack’ the structure of a problem” (2008). Visualizations help students to think and process mathematical ideas when solving problems and therefore; learn with understanding rather than simply mimicking teacher practices and procedures.

Edens and Potter cite the work of Mayer, Sims, and Tajika (1995) and Mayer (2003) noting, “Mathematics textbooks and tests emphasize procedural knowledge where students engage in drill and practice exercises carrying out computational procedures […] Systematic instruction on how to execute strategies for translating problems and making meaningful representations of problems, however, is not always given” (2008). This ties into the idea of traditional and reform teaching methods and the need for teaching and learning with understanding. As technology advances, it makes less sense for students to focus on computation when calculators are readily accessible. Incorporating instructional technology in the classroom can maximize the development of critical thinking, reasoning, and problem solving skills because less time is wasted on computational procedures. Computational skills have become less relevant in the 21st century.

Considering the work of Van Essen & Hamaker (1990), Edens & Potter state, “[Visualizations can be] approached in two different ways: (1) teaching students specific types of diagrams or (2) instructing students to make their own diagrams” (2008). Participants will explore these two methods during this course.

**Goals**

Participants will develop their skills to recognize and create quality visualizations to teach mathematics with understanding.

**Performance Objectives**

- 1. Participants will
**compare and contrast**reform teaching methods, methods that encourage teaching for understanding, and traditional teaching methods.

- 2. Participants will
**identify**advantages of using visualizations for teaching and learning mathematics with understanding.

- 3. Participants will
**identify**characteristics of meaningful visualizations for supplementing lessons with understanding and**develop strategies**to create useful visualizations for this purpose.

- 4. Participants will
**analyze**a series of visualizations created with Geometer's Sketchpad, GeoGebra, and recorded teacher lessons, to model strategies for designing classroom lessons and/or materials.

- 5. Participants will be able to
**construct**a visualization for a given mathematical concept.

- 6. Participants with
**reflect**on ways to transfer skills for using visualizations in mathematics to students by collaborating in a peer discussion.

**Task Analysis**

**Prerequisites:**

- Participants should have a general understanding of how students learn and differentiated instruction.
- Participants should have experience using and creating visual aids.
- Participants should have experience planning lessons.

**Unit 1: What do we mean by “Learning with Understanding”?**

- Participants will explain the advantages of learning with understanding by completing a two paragraph refection.
- Participants will compare reform teaching methods and traditional teaching methods by analyzing research.
- Participants will clarify any misconceptions about learning with understanding in a class discussion.

**Unit 2: Creating Meaningful Visualizations for Teaching Mathematics**

- Participants will identify how visualizations supplement learning with understanding by collecting and analyzing research.
- Participants will explain if and/or when visualizations are useful tools for students.
- Participants will clarify any misconceptions about the importance of visualizations in a class discussion.

**Unit 3: Putting Ideas into Practice**

- Participants will analyze a series of visualizations created with Geometer's Sketchpad to model strategies for designing classroom lessons and/or materials for using visualizations to teach mathematics with understanding.
- Participants will analyze lessons created by teachers and write a reflection on how they integrated visualization into their lessons to teach mathematics with understanding.
- Participants will construct a visualization.
- Participants will create a lesson with visualizations.
- Participants will critique lessons created by their peers.
- Participants will reflect on ways to transfer skills for creating visualizations to learn with understanding to students using a class discussion.

**Curriculum Map**

**References and Resources**

- Anderson-Pence, K. L., Moyer-Packenham, P. p., Westenskow, A. a., Shumway, J. j., & Jordan, K. k. (2014). Relationships Between Visual Static Models and Students' Written Solutions to Fraction Tasks. International Journal For Mathematics Teaching & Learning, 1-18.

- Carpenter, T. and R. Lehrer. (1999). Chapter 2: "Teaching and Learning Mathematics with Understanding." From:
*Classrooms That Promote Understanding*edited by Elizabeth Fennema and Thomas A. Romberg. Mahwah, NJ: Erlbaum. (Accessed via books.google.com). pp 19-32.

- Common Core State Standards. http://www.corestandards.org/Math/

- Dimmel, J. and P. Herbst. (2015). "Semiotic Structure of Geometry Diagrams." Journal for Research in Mathematics Education. Volume 26. Number 2.

- Edens K, Potter E. How Students "Unpack" the Structure of a Word Problem: Graphic Representations and Problem Solving. School Science And Mathematics [serial online]. May 1, 2008;108(5):184-196. Available from: ERIC, Ipswich, MA. Accessed February 23, 2015.

- Engage NY. Grade 6 Mathematics Curriculum: Module 1, Topic A, Lesson 3. Accessed May 8, 2015 via: https://www.engageny.org/resource/grade-6-mathematics-module-1-topic-lesson-3

- Geometer's Sketchpad: https://www.mheonline.com/programMHID/view/00000SPAD/

- GeoGebra: https://www.geogebra.org

- Goldin, G. A. (2002). Representation in mathematical learning and problem solving. Handbook of International Research in Mathematics Education. March 1, 2002. Available from: Routledge, New York, NY. Retrieved from: https://books.google.com/books?id=LDGRAgAAQBAJ&dq=Goldin,+G.+A.+(2002).+Representation+in+mathematical+learning+and+problem+solving.&lr=&source=gbs_navlinks_s Accessed February 23, 2015.

- Guzman, M. The Role of Visualization In the Teaching and Learning of Mathematical Analysis. Universidad Complutense de Madrid. Accessed May 8, 2015 via http://www.math.uoc.gr/~ictm2/Proceedings/invGuz.pdf

- "How Math Comes to Mind: Intuition, Visualization, and Teaching." Princeton University. Accessed May 13, 2015 via: https://mediacentral.princeton.edu/media/How+Math+Comes+to+MindA+Intuition,+Visualization,+and+Teaching/1_0bfnmopc

- Teacher Channel:

- You Tube:

**Links**

**Mini-Course:** Using Visualizations to Teach Mathematics with Understanding (Spring 2015 Mini-Course)

**About Me:** Melissa Connor