Introduction
Advanced Pre-Calculus is a fourth year (senior level) course. A senior is expected to take “some intentional responsibility for his own growth. He strives to grow in the realization and use of his talents… [He] is learning how to learn; consequently, he desires learning as a life-long process to develop imagination, feelings, conscience, and intellect. He is developing compassion for others by… accepting and offering constructive criticism. [He] is willing to risk new experiences and recognizes them as opportunities to grow…” The student is “encouraged to exhibit a mastery of academic skills for advanced education and development of character. He is [being] trained… in basic study and research skills so that he is prepared for life-long creative learning. He is developing an aesthetic capacity. [He] is developing intellectual integrity and honesty. His repertory of knowledge enables him to view his community in a compassionate and hopeful way. Similarly, he is developing a critical consciousness which enables him to analyze and question the issues facing our society. He is learning that different branches of knowledge provide alternate approaches to truth” (MUHS 2003).
Course Description
This course is the final preparation before taking Calculus. Topics in Algebra are interrelated with exponential, logarithmic, and polar functions, with a heavy concentration on trigonometric functions. Sequences, series, proof by induction, and limits of functions and sequences are covered along with detailed analysis of various types of functions. This course moves at a more rapid pace and covers more topics in-depth than Precalculus. (MUHS 2007)
We will focus on families of functions and their graphs during the first semester. We will expand our knowledge of trigonometry during the second and third quarters. In the fourth quarter, we will explore topics including matrices, series, polar coordinates, and conics (MUHS 2003). Much of the content of this course has been introduced or covered in prior courses. This course emphasizes increasing a depth of that knowledge, and making connections between concepts. Specifically, students are challenged to strengthen their understanding analytically and visually, algebraically and graphically.
Course Goals and Objectives
The larger course goals are for you to further develop the following skills and abilities: problem solving, critical thinking, learning to learn, perseverance, time management, and organization. These goals do not reflect math skills but life skills. These skills and abilities are applicable in every aspect of life, as well as every course you will take in your academic career.
This project has two parts: a portfolio of work demonstrates a student's problem solving strategies while the performance assessment provides the opportunity to demonstrate one of those strategies to the class. The majority of the work will be completed outside of class. The assessment (both the portfolio and performance assessment) will be assessed on the student’s ability to select strategies to solve non-routine problems as well as his ability to communicate his problem solving processes effectively in both oral and written formats; the performance assessment addresses the oral aspect of the assessment while the portfolio addresses the written.
Outcome(s)
Our
Jesuit Identity: “Since 1857,
Learning experiences that provide skills as well as enduring knowledge that transcends the curriculum and encourage life-long learning and maximize student potential (Wiggins & McTighe 2005). Additional research, such as Thoughtful Education Press 2005, posits four areas of academic literacy:
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· Collect and organize ideas via note-making · Make sense of abstract academic vocabulary · Read and interpret visual displays of information |
Reflective skills · Construct plans to address questions and tasks · Use criteria and guidelines to evaluate work in progress · Control or alter mood impulsivity |
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Thinking Skills · Draw conclusions, make inferences, hypotheses, conjectures, and test them · Conduct comparisons using criteria · Analyze the demands of different kinds of higher-order thinking questions |
Communication skills · Write clear, well-formed, coherent explanations in all content areas · Write comfortably in non-fiction genres: problem/solution, decision-making, argument, comparative · Read and write about 2 or more documents |
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© The Thoughtful Education Press, 2005. From Teaching for Rigor, Thought & Authenticity. Silver, Strong & Assoc. (2007) |
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Mathematics Outcomes
With emphasis on enhancing openness to growth and intellectual competence, this project is designed to also align with the Mathematical Processes Content Standard from the Wisconsin Model Academic Standards for Mathematics (WMAS): “Students in Wisconsin will draw on broad body of mathematical knowledge and apply a variety of mathematical skills and strategies, including reasoning, oral and written communication, and the use of appropriate technology, when solving mathematical, real-world* and non-routine* problems” (WMAS 1998). The rationale for this standard is stated as, “In order to participate full as a citizen and a worker in our contemporary world, a person should be mathematically powerful. Mathematical power is the ability to explore, to conjecture, to reason logically and to apply a wide [repertory] of methods to solve problems. Because no one lives and works in isolation, it is also important to have the ability to communicate mathematical ideas clearly and effectively [in writing and in speech]” (WMAS 1998).
Purpose
In the portfolio, each student will showcase three recreational problem solving exercises he has successfully completed. This will allow the student to review his problem solving strategies and to self assess his problem solving ability and development, as well as his ability to communicate his ability in a written format. The portfolio provides an opportunity for the student to develop and demonstrate his proficiency with integrating digital images and word processing software.
In the performance assessment, each student will present one recreational problem solving exercise, which may or may not be included in the portfolio, to the class. Each presentation must focus on a unique exercise. This oral presentation will allow the student to the opportunity to communicate verbally to a group, and provide an opportunity to self assess his problem solving communication skills when he reviews his video taped performance.
Directions
Academic Honesty Constraints for This Project
YOU MAY talk through exercises with others,
however you must include who you worked with and describe the extent in the
process section for the puzzle. For example: A student worked on a specific
puzzle for over 2 hours and he "just couldn't get it." He reported
that he used many different problem solving strategies. He noted that he
thought about it, thus reflected. He wrote that he drew pictures or sketches.
He described how he used pencils to construct a model (or physical
representation) of the puzzle. He also wrote that his 4-year-old sister kept
asking him what he was doing. He initially replied "my homework" and
asked her to be patient. After numerous interruptions, he explained that he was
trying to make the pencils look like the picture and explained what the
constraints (or limits in the directions) were. After explaining what he was
trying to do, she then moved the pencils and solved the puzzle. He wrote that
he was very frustrated and irritated because he was a senior in high school
that spent hours on this puzzle and his 4-year-old sister solved it right away.
Note 1: Make certain that you praise and thank those that help you. Note 2:
This project aims to reinforce the values of Jesuit Education. You will need to
connect your experience(s) to one or more characteristics of the graduate at
graduation. In the above example, the student was challenged to be open to
growth and learn from his little sister. He was reminded that everyone has
different gifts. Note 3: A learned man
acquires knowledge. An intelligent man applies his (formally and informally)
acquired knowledge. Thus, an intelligent man transfers skills and abilities to
new challenges and experiences to become successful. YOU MAY NOT have someone
else do the puzzle(s) for you and submit his/her work as your own.
YOU MAY ask someone for help with scanning,
taking digital photos, or word processing. YOU MAY NOT have someone scan, type,
format or otherwise do the work for you.
Portfolio
The student will select three exercises to showcase in the portfolio. The goal is to demonstrate a variety of problem solving strategies, as supported by the overall portfolio criteria. Remember that a secondary objective of the portfolio is demonstration of technological competency. Specifically you will be assessed on your ability to efficiently scan images, word process, and integrate digital images into a single word-processed document that is formatted MLA-style.
Sample
Format of the Portfolio
Use Subheadings to organize your paper. You may wish to change the font of the headings/subheadings, and some creative expression is fine here. Please be certain that the font that you select is readable, and doesn’t take up the whole page. (Refer to the sample above.)
Puzzle 1, 2 or 3 (Heading
1: centered)
Place the scanned image(s) of the puzzle, including the puzzle number and title, that you selected. Note that scanned images for printing are typically between 200 and 300 dpi. Scan as grayscale (not color) to reduce the image's file size.
Process (Heading 2: left flush)
Under the process subheading, explain – in detail – your process for selecting and solving the exercise. If you liked the puzzle because the image initially caught your attention, then say so. For example: “I chose this puzzle because it had an interesting picture that caught my eye. I initially read the moral of the puzzle, then I read it from the beginning. I reread it, focusing on the part where numbers are presented. “ Continue to write about how you approached the puzzle. “I thought about” it is NOT sufficient. What did you think about? Did you picture the money transactions and the kid crossing the bridge? If so, then you used visual images. Did you draw pictures, or write it out? Whatever approach you use is perfectly fine, however you MUST explain and support your reasoning.
If you draw pictures or sketch equations, save you work and scan it. It doesn’t matter if the scribbles are neat; it matters that you are providing specific evidence of your problem-solving process. The “Process” portion of the puzzle is typically one page long. This section must be double spaced, 12 point Times or Times New Roman.
Solution (Another heading 2: left flush)
This section must also be double spaced, 12 point
Times or Times New Roman. This is where you explain your solution in DETAIL.
You may use words, symbols, tables, drawings, etc., however all of the work
must be contained in this one word processed document; it must be all digital
and integrated into one word document. You will be handing it in
electronically, so it will be assessed on this as well. If you need help with
this, please let me know sooner rather than later.
All images or written work must be
scanned. All algebraic equations can be either typed or scanned in. I
can help you with typing math equations in MSWord. I would like to help you
with this, however I will not help you the day before or the day that it is
due. You must PLAN and manage your time!
Performance Assessment
For this performance assessment, you must select one of the recreational exercises from the provided packet to present to the class. The exercise that you present may or may not be one of the exercises included in your portfolio. However, only one student may present an exercise. Each student will present a unique exercise. Therefore, begin your work early enough that you will be able to sign up as soon as possible. The sign up sheet will be posted on the bulletin board in our classroom. Use ink to sign up. Once someone has signed up, that exercise has been selected; you must choose a different exercise to present. (If you are caught altering the sign up sheet, severe disciplinary action will be taken.)
You have 3 to 5 minutes to demonstrate a problem solving strategy that actually solves a problem. (Yes, you have to solve the exercise you select.) Remember that an objective for this project is to “…draw on broad body of mathematical knowledge and apply a variety of mathematical skills and strategies, including reasoning, oral and written communication, and the use of appropriate technology, when solving mathematical, real-world* and non-routine* problems” (WMAS 1998).
This presentation will be video taped. Relax, you will be the only one to view this tape, unless you let someone else see it. (I won’t even watch it.) The intent is to provide an opportunity for you to view yourself when you speak so that you can accurately self assess and improve. This allows you to be an audience member of your own presentation. Therefore, you must provide a VHS videotape to your teacher. Failure to give this tape to your teacher on time will result in nightly JUGs until the tape is received. In addition, failure to provide and view the tape will result in a zero (0) grade on the self assessment, which will negatively effect your grade.
Criteria
Your presentation will be assessed on your ability to communicate your problem solving strategy or strategies that aided you in finding a solution or solutions to the exercise that you selected. This includes your ability to communicate or present well. The following pages note MUHS’ Guidelines for Speakers. Don’t let all of this overwhelm you; have fun with it. The best presentations for this project have been relaxed, fun, and engaging. Yes, the audience has been a part of prior presentations, and it is a great speaking practice to involve the audience. (Would you rather sit and listen, or do something?)
Remember, you are with your class; you know these guys. (And remember that they have to do this too.) Relax and have fun. Props are good, so are handouts. However, please do not print handouts in the computer labs. (Multiple printings are a violation of the school’s computer usage agreement that you signed at the beginning of the year.) Give me one clean original, on white paper, along with the number of copies you want and on what color paper, etc., and I will photocopy the handouts for you. If you want me to copy your handouts, you must have the original(s) and photocopying instructions.
Timeline
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What is due… |
on or before… |
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1st digital
puzzle |
Monday October 29 |
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1st & 2nd
digital puzzle |
Monday November 5 |
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VHS video tape (for 3 -5
minute speech) |
Wednesday November 7 |
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Any handouts you want copied for the presentation,
along with printing instructions (single or double sided, number of copies) |
Monday November 12 |
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Complete project with
title page (hard copy and electronic
copy) |
Monday November 12 |
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Presentations |
Friday November 16 –
Tuesday November 20 |
Assessment Rubrics
Both the portfolio and the presentation will be assessed by the student, three of his peers/classmates, and the teacher. The assessment will be based on the criteria on the provided rubrics.
Rubric A, parts 1 and 2, for the performance assessment, will be completed based on the performance assessment by the student, three peers and the teacher. The self assessment must be completed and handed in before the peer and teacher feedback will be made available to the student. The self-assessment includes the completed rubric as well as your word processed response to the following:
· What did you like about what you did?
· What would you do differently next time?
This rubric is based on and dependent upon the MUHS Guidelines for Speakers (included). If your name is called and you are not ready to present, you will be called first on the following day; your grade will be reduced one full letter grade each day you are not prepared when called to present.
Rubric B, the portfolio individual exercise assessment rubric, will be completed by the student, three peers and the teacher. This rubric will be completed by these individuals for each of the five required exercises. All five exercises will represent 50% of the portfolio grade; each exercise is 10% of the overall grade.
Rubric C, for the overall portfolio, will be completed by the teacher and the author of the portfolio. It will be weighed for 50% of the portfolio grade.
A copy of each rubric is contained on the following pages. Please read through each one, so that you understand how your grade will be determined before you begin you work.
Works Consulted
Herr, T. & Johnson, K. (1994). Problem
solving strategies: crossing the river with dogs and other mathematical
adventures.
JSEA
(2007) What makes a Jesuit school Jesuit?
Kolvenbach, Father General, R.P. Peter-Hans, S.J.
(December 8, 1986). The Characteristics
of Jesuit Education. Presents at the International
Commission on the Apostolate of Jesuit Education (ICAJE) in
National Council of Teachers of Mathematics.
(2004) Principles and Standards for
School Mathematics. Retrieved
Stiggins, R. J. (2001) Student-involved Classroom Assessment.
Wiggins, G. & McTighe, J. (2001). Understanding by Design.
Wisconsin Department of Public Instruction. (2007).