Resource Guide: May 2012

ORGANIZING TOOL

This guide contains the following sections:

                             I.                 INTRODUCTION
   II.    LESSON PLANS
   III. RESOURCE LINKS

Click on youtube.com/ferrantemath

I. INTRODUCTION

Genuine constructivist learning strives to create an authentic learning environment from which learners can do the following: (1) articulate and access prior knowledge on a related and previously unrelated subject areas; (2) allow users to experiment, conduct inquiry, thereby creating new schema; (3) support the possibility of cooperative learning; (4) allow users to play as a means of practicing real world experiences; (5) take the emphasis off expert teaching and allow for novice-centered learning; and (6) create a virtual environment that is genuine and authentic (Shore, 2012).

As a math instructor for the last twenty-three years, I previously believed that I became a better instructor as the years progressed because I became more comfortable with my subject area. In some respects, I still believe that is true. Upon reflection however, I think my pedagogy may have changed primarily because years ago, I made a commitment to enter into the math experience from the point of view of my students, to try to experience learning algebra again through the eyes of a teenager -- no small task for an adult! What crystalized for me in the last few years is that learners learn more effectively by DOING not necessarily by being lectured to . . . and hence my turn toward Constructivism and constructivist projects. In the Exploratorium in San Francisco, there is a message board that reads:

People retain

10% of what they read

20% of what they hear

30% of what they see

70% of what they talk over with others

80% of what they use and do in real life

95% of what they teach to someone else.

A perfect capsualization of constructivist learning!

In constructing this guide, I learned that over the years, I inadvertantly developed constructivist activities without realizing there is an educational theory called Constructivism. This project (creating this resource) has given me to opportunity to re-create old lesson plans and activities with a more focused constructivist intention. As indicated in the opening paragraph, there are six characteristics of genuine constructivist learning. I have been able to recreate a number of previous activities keeping in mind the balance between the six different characteristics. For example, I found that in many of the activities I only cursorily paid attention to the reflection component of a genuine constructivist learning. In recreating one of my more successful lessons -- discovering parabolas in art and architecture -- I added a reflection component before and after the lesson. As indicated by a number of constructivist theorists, the reflection component helps to reinforce and strengthen prior knowledge and helps to create new links between different concepts (Okebukola, 1992).

My plan going forward is to continue to create new activities and projects using constructivist principals and using this site as a repository. At the same time, I will continue to reexamine old lesson plans and recreate them using a more constructivist approach.

References

Okebukola, P. A. (1992). Attitude of teachers toward concept mapping and diagramming as

metalearning tools in science and mathematics. Educational Research, 34(3), 201-213.

Shore, L. (2012). Authentic learning environments, simulations, and digital media tools.

(PowerPoint presentation). Constructivism and Technology. University of San Francisco, Spring.

II. LESSON PLANS

The Parabola Project: Connecting Art, Architecture and Mathematics is an example of a new lesson plan born out of the fruits of a Constructivist mentality. The set of plans brings together three of my favorite subject areas: art, architecture, and mathematics. The project can be substituted for an entire chapter in a traditional math text that covers quadratic equations. The plans can be lengthened to four or five days depending what new extensions can be developed by the students or the instructor.

The Parabola Project:

Connecting Art, Architecture and Mathematics

LESSON 1: Recognizing Parabolic Form in Art and Architecture

Duration: 50 minutes

Intended Audience/Grade Level

Audience: High School Algebra Faculty
Mathematics Level: Intermediate Algebra (or accelerated Elementary Algebra)

Misconception Addressed by Proposed Activities

Parabolas are confined to mathematics textbooks and have no real-life applications.

Learning Objectives

· Students will visually recognize parabolic forms in photographs of art and architectural landmarks.

· Students will practice and refine internet-based research methodologies.

· Students will conceptually connect quadratic equations with applications of architecture.

· Students will develop an increased appreciation of the utility of algebra in the world around them.

Materials

teacher computer
projector
internet access
graph paper
student computers/iPads

Assumptions/Prerequisite Skills

Students have completed introductory linear graphing projects.
Students are able to recognize the Standard Form of a quadratic equation: y = ax² + bx + c
Students know that a quadratic equation produces the graph of a parabola.
Students are members of small groups for ongoing group activities.

Activity #1: “What do you know about parabolas?” (10 minutes)

Part A (Establishing the Misconception):

As class begins, students are seated in their established small groups (three to four members). The instructor begins the class by asking: “What do you know about parabolas?” The instructor should write down all germane responses on the overhead projector or the whiteboard.

Many students will give mathematical answers such as “They are quadratic equations” or “They have an x-squared.” The instructor should note all these responses on the overhead without comment. Other students may provide real-world examples “They are the MacDonald Arches” or “They are used in bridges.” Again, note all responses on the overhead.

In practice, students are very enthusiastic about showing off their knowledge of parabolas, however many of their responses reveal a fragmented and isolated set of observations. In educational parlance, many of the responses reveal incomplete or poorly formed schema; “holes” in an overall understanding of how the different concepts relate to one another. If possible, the instructor should save responding to these misconceptions in the hope that the students come to correct their deficiencies by constructivist methods.

Activity #2: Examples of Parabolas in Architecture – Student Suggestions (15 minutes)

After student discussion of the responses in Activity 1, the teacher should guide the discussion by asking students for examples of parabolas appearing in architecture.

Students will typically answer “McDonalds”, “Golden Gate Bridge”, “St. Louis Arch” or “Roman Aqueduct”. As students call out suggestions, the teacher uses the teacher’s computer and projector to do quick internet searches for the suggestions made by students, and points out the parabolic forms in the suggestions made by students.

The Roman Aqueduct is a popular response.

Activity #3 Examples of Parabolas in Architecture – Teacher Guided Discussion

After student discussion of the responses in Activity 2, the teacher should extend the lesson by showing students other examples of parabolas appearing in architecture.

Useful examples from modern architecture appear on Santiago Calatrava’s website. Calatrava is a modern-day Spanish architect whose designs show beautiful integration of geometric form , sculpture, and architecture:

http://www.calatrava.com/

Other “show and tell” examples that have been used successfully with students are the Duomo in Florence, Italy and the Roman Coliseum

Activity #4 Research and Identification of Parabolas (35 minutes)

After identifying parabolas and other geometric forms in multiple examples of architecture, students are given the mission to search the internet for interesting examples of parabolic form. If students have classroom access to iPads, computers, and or printers, tell them to do the following:

1) Find a good example of parabolas used in art or architecture.

2) Print the example. Try to make the printout of the picture rather large – almost taking up a full sheet of paper if possible.

3) Find two or three examples. Instruct the students that examples with parabolas that are not “slanted” work best.

4) Have graph paper available for students to trace the parabola onto a graph.

5) Whatever is not accomplished in class can be assigned for homework that evening.

The Parabola Project:

Connecting Art, Architecture and Mathematics

LESSON 2: Creating Sketches and Developing Equations for Parabolas

Duration: 50 minutes

Intended Audience/Grade Level

Audience: High School Intermediate Algebra Teachers
Mathematics Level: Intermediate Algebra (or accelerated Elementary Algebra)

Misconception Addressed by Proposed Activities

Parabolas are confined to mathematics textbooks and have no real-life applications.

Learning Objectives

· Students will visually recognize parabolic forms in photographs of art and architectural landmarks.

· Students will practice and refine internet-based research methodologies.

· Students will conceptually connect quadratic equations with applications of architecture.

· Students will develop an increased appreciation of the utility of algebra in the world around them.

· Students will identify components and characteristics of a quadratic equation, including coefficients (a,b,c), dependent variable, independent variable, quadratic terms, linear terms, constant terms, standard form, x-intercepts, y-intercepts, vertex, and axis of symmetry.

· Students will identify three data points and create a 3x3 system of equations.

· Students will solve a 3x3 system of equations without use of a graphing calculator.

· Students will increase fluency in the language of mathematics by writing quadratic forms describing real-world architecturally-based parabolic forms.

· Students will develop a sense of empowerment and self-efficacy in their ability to creatively construct mathematical representations.

· Students will develop stronger analytic skills, including identification of underlying constructs central to the problem, sourcing information necessary for procedural accuracy, and learning through inquiry.

Materials

computer and projector
printer
graph paper

Assumptions/Prerequisite Skills

Students have completed introductory linear graphing projects.

Students are able to recognize the standard form for a quadratic equation y= ax² + bx + c will produce the graph of a parabola.

Students know how to solve a 3x3 system of equations

Students are members of small groups for ongoing group activities.

Activity #1 Where is Your Parabola? (5 minutes)

Part A (Review of first lesson and extension to computational mathematics):

As in Lesson One, as students arrive, have them sit in their respective groups. On the overhead projector is the questions: “Where is your parabola project at this moment?” As before, write down the responses

They are asked to show each other the photographs that they brought to class, and discuss locations and the appearance or patterns that they can identify.

Activity #2 Sketching the Graphs of Parabolas in Architecture (15 minutes)

After student discussion of the responses in Activity 1, the teacher should guide the discussion toward the mathematical computational exercises.

First, students are asked to trace the parabola on graph paper. Students may darken the outline of the parabola on their internet picture by using a black marker and/or hold the graph paper up to the window so that the shape of the parabola can be more easily traced on the graph paper.

The teacher should give the following set of instructions:

1.) Start by sketching the x-axis and y-axis on a sheet of graph paper.

2.) Lay your graph paper over the picture of your building, and trace the shape of the parabola onto your graph paper.

3.) Eventually, you will need to identify three points. It will be easier to do this if you try the following:

a.) Make sure your parabola intersects the y-axis in one point.

b.) Make sure your parabola intersects the x-axis in two points.

c.) Try to orient your parabola so that it intersects the “cross-hairs” of the graph paper, so that you are dealing with whole numbers and not fractions when you select points. (All students hate generating equations from fractional coordinates, so there is no difficulty in understanding this instruction.)

After the parabolas have been transferred to graph paper, students are asked to identify three points on the graph. Students in groups collaborate and communicate to confirm each other’s accurate identification of

points.

Activity #3: Creating the Equations of Parabolas in Architecture (30 minutes) although time can be adjusted according to the progress of the group)

After groups have successfully identified and verified the coordinates for the 3-4 examples per group, the most difficult part of the activity begins. Each group must use the (x,y) coordinates of the three points to set up a 3x3 system and solve for the coefficients of a, b, and c. FerranteMath on YouTube has a video that is instructive here.

http://www.youtube.com/ferrantemath

Go to the website and type in “Finding a quadratic equation using three data points” into the search bar on the website. The results will bring you immediately to the videos.

As an alternative, the teacher can instruct the class in the following manner:

Using the students’ coordinates of (-13,0), (0,9), and (14,0) the group would set up and solve the following 3x3 system:

Standard Form of a Quadratic Equation: y = ax^2 + bx + c

Start the process by substituting the x and y coordinates into the Standard Form of the Quadratic Equation.

For the first coordinate: (-13, 0) i. 0 = a(-13)^2 + (-13)b + c

ii. 0 = 169a -13b + c

For the second coordinate: (0,9) i. 9 = a(0)^2 + (0)b + c

ii. 9 = c

For the third coordinate: (14, 0) i. 0 = a(14)^2 + b(14) + c

ii. 0 = 196a + 14b + c

Simplify each equation and now the students have established a 3x3 system of equations! At this point, a review of how to solve a 3x3 system might be appropriate. Tell students they can use youtube.com/ferrantemath for help if they need review on how to solve a 3x3 system manually.

In reality, this part of the exercise can be quite frustrating for students. Allow plenty of time for groups to solve their 3-4 problems, typically two class periods.

After two days of group work, and moments of extreme struggle and frustration, students were able to arrive at solutions and generate their parabolic equations. It is amazing to see the frustration evolve into excitement and heady feelings of success!

Whatever is not accomplished in class can be assigned for homework. Again remind the students that help in establishing a 3x3 system of equations is available in the FerranteMath YouTube site.

The Parabola Project:

Connecting Art, Architecture and Mathematics

LESSON 3:

Confirming Invented Equations Through the Use of Graphing Software

Duration: 50 minutes

Intended Audience/Grade Level

Audience: High School Mathematics Faculty

Mathematics Level: Intermediate Algebra (and accelerated Elementary Algebra)

Misconception Addressed by Proposed Activities

Parabolas are abstract mathematical constructs with no real-life applications.

Learning Objectives

· Students will learn how to use graphing software (apps or programs) to confirm computational solutions for invented quadratic equations.