Math I: Gretchen Verner

Assignments                                 Answer Keys                               Student Work       

Math I - 2007-2008 - Gretchen Verner

Students come into the three sections of Math I with a wide range of mathematical backgrounds, so forming a cohesive group of learners and introducing them to what is, for some, a very different way of learning mathematics is a challenge.  The primary goals of this first course in CPS's integrated math sequence are

• to increase students' appreciation for, and interest in, mathematics

• to help students become more confident problem-solvers and learners of technical material.

An important byproduct of achieving these goals is providing students with a solid foundation for four years of high school mathematics and science.  There are several basic principles that govern our approach:

•  retention of basic concepts, skills and techniques is necessary for understanding more complex ones

•  students will remember what makes sense to them

•  difficult ideas are mastered best through continuous reinforcement over time

•  robust student learning is enhanced when students engage in mathematical conversations with their peers (facilitated by grouping seats so small groups of students can easily work together)

•   mathematical learning must incorporate problem-solving

For this approach to work, lessons must be carefully designed to guide students to the important ideas, facts and algorithms of the current topic and help them make connections between topics.  These lessons gradually build a solid trunk of basic understanding that will support smaller branches containing the many facts and procedures they must master.

Mathematical Content of Math I:

Algebra: Many Math I students have had a full year of algebra in middle school, so we review and deepen understanding from novel perspectives whenever possible.  We encourage depth of understanding by asking students to communicate their method clearly, both orally and in writing.  We return to first principles often to reinforce and emphasize them.  For example our work with fractions emphasizes multiplication by a form of one as a way to make a fraction look different without changing it, and our "solve or simplify" problems require that students first determine whether the collection of symbols is an equation to be solved or an expression to be simplified.  Use of a scientific calculator is taught explicitly, and students learn to recognize the calculator as a valuable investigative tool.  But powerful tools make for powerful mistakes, so mental arithmetic, mental algebra, and "no calculator" questions help guard against excessive calculator dependence.  As mathematical software develops as an effective teaching tool, we use it more frequently.  We explicitly teach checking skills, give students common errors to analyze, and strongly encourage them to test the reasonableness of their results.

Geometry: We teach vocabulary and basic facts for angles, polygons, circles and solid figures.  Geometric transformations such as reflection and rotation, often in a coordinate context, are explicitly taught and used as a theme throughout CPS math.  Rigorous proof is postponed to Math II, but an introduction to mathematically precise definitions and some justification of basic theorems is included.  Students make extensive use of physical models of geometric figures and occasionally use dynamic geometry software in a guided discovery environment.

Probability and Statistics: Students learn how to use counting principles to determine the number of possibilities for a given situation.  They use this information to compute simple probabilities.  Simulations illustrate the difference between theoretical and experimental probability.  Basic statistical ideas are introduced or reviewed through activities and carefully designed examples.

We coordinate closely with Physical Science teachers to reinforce skills that students will need in that course, such as scientific notation, conversion factors, and algebraic manipulation of literal equations.

COURSE OUTLINE, not in chronological order.  We cover several topics simultaneously.

Past and present CPS math teachers wrote the text and handouts for this course.  We started using our own materials in the 1980's and have revised them most summers to correct errors, reflect student feedback and keep the content up-to-date.  We started doing this because standard texts were inadequate for the following reasons, among others:

≤    latest research on learning math not utilized

≤    topics presented in isolation without taking advantage of connections between them

≤    too much emphasis on "how" and not enough on "why"

≤    too few questions designed to deepen understanding

≤    mathematical jargon and notation introduced too quickly

≤    insufficient emphasis on problem-solving and conceptual understanding

≤    typical student errors not discussed sufficiently

≤    standard textbooks heavy and expensive, with no room to solve problems right in the text

≤    not tailored to the high ability and motivation of CPS students

≤    not revised annually

Certainly there are disadvantages to using out own materials, but we do our best to minimize them.  For example, this year we are making more support materials and extra practice available for students.

We use the following grading rubric in Math I:

20% – Group and class participation. 

20% – Homework, graded reviews (colored sheets) and Problem Sets.

20% – Quizzes

25% – Tests

15% – Semester or final exam

We need to accommodate a wide variety of skill and interest levels, especially at the beginning of this course.  We expect students who need more help or explanation than we can provide in class to make an appointment with their teacher.  The resulting plan for improvement may include regular visits to the Learning Center or tutoring sessions with a suitable CPS student.  Students seeking an extra challenge can participate in math club or math contests, and they should talk to me about their situation.

If you want more information on our program, please contact me: gretchenv@college-prep.org.