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Development of Mathematics Expectations Development of Mathematics Expectations

The PK-16 Mathematics Advisory Committee is a working committee of the PK-16 Council
chaired by Governor Donald Carcieri. The committee formed in March 2004 has met and
communicated since that time to develop guidelines for college readiness in mathematics. The
committee was chaired by Prof. Lewis Pakula of URI and co-chaired by Judith Keeley of RIDE,
with representatives from RIC, CCRI, RIDE, and a number of Rhode Island high schools. (See
full list below.) The committee agreed on the following principles about students expectations,
excerpted from the Executive Summary of the committee's final report:

All students who expect to take and succeed in entry-level courses at the university/college level
should be proficient in what was once understood as Algebra II. In particular, they should be able
to solve problems in arithmetic and algebra, particularly those involving fractions in any form,
using conventional notation and algorithms. Such problems should entail appropriate and correct
use of arithmetic and algebraic methods and reasoning, and the correct systematic computation
of the answer. Students should be able to do small-scale problems without use of a calculator and
be able to make appropriate use of a calculator and other technology in applications. Students
should have the widest possible experience solving problems involving application of
mathematics, as well as problems that call for careful and extended reasoning about mathematics
itself.

All students should have facility with geometric reasoning. Among other things, they should
know the formulas for area, perimeter and volume related to basic geometric objects and be able
to use these to solve problems involving more complicated objects. They should be able to
reason about similar and congruent triangles and know the Pythagorean Theorem, its
applications and related trigonometry, and, ideally, a proof. They should be able to form
inverses, converses and contrapositives of geometric (and more general) statements, give
definitions of geometric terms, and make geometric deductions.

Topics in discrete mathematics and probability that reinforce basic competence in algebra and
arithmetic are valuable for students intending to take general education courses at the college
level.

While geometry, statistics, discrete mathematics, and other fields are obviously important, the
strong consensus among mathematicians at the college level is that inadequate mastery of
arithmetic and algebra is the greatest impediment to success in college mathematics, whether for
general education or in mathematically intensive courses of study. Those curricular reforms in
recent years that encourage more problem solving, student engagement, and lively content are
laudable and should complement the rigorous development of the basics for college readiness.

These content expectations are intended to be consistent with the RI GSE's 11-12 as developed.
Further details will be found below.

PK-16 Mathematics Advisory Committee
Joseph Allen, Department of Mathematics, CCRI
Peter Andreozzi, RIDE Donna Christy, Department of Mathematics, RIC
Patricia Dulac, East Greenwich High School
Barbara Fox, RIDE
David Heskett, Department of Physics, URI
Judith Keeley, RIDE, Co-chair
Paula Najarian, Tolman High School
Ann Moskol, Department of Mathematics, RIC
Lewis Pakula, Department of Mathematics, URI, Chair
Diane Schaefer, RIDE
Lois Short, Burrillville High School

Mathematics Readiness without Remediation

Students should enter college prepared for success and equipped to make choices of major and
career based on their interests, talents, and aspirations, and not limited by their mathematics
background. For example, a student who may need only a single finite mathematics course for
her initial major choice in psychology should be able to switch to a precalculus/calculus track if
her interests change to marine biology, without remedial coursework. Thus, these
recommendations apply, in principle, to all students entering four year college programs or who
intend to transfer to such programs from a community college.

For example, both URI and RIC have stated admissions requirements in mathematics of one year
of geometry and two years of algebra. Interpretation of these requirements, consistent with their
original intent and current expectations, is indicated below. Although students in general
education mathematics courses might make active use of a different subset of the items below
than those taking technically oriented courses, the core arithmetic/algebraic skills, and
experiences with mathematical reasoning, problem solving and presentation, are relevant to all
students.

There is wide agreement that an informed fluency with the algorithms and methods of arithmetic
and basic algebra, based on principles and reasoning, is a cognitive and conceptual precursor to
more advanced mathematics.

Basic Skill and Knowledge Expectations

Students should

Solve a wide variety of problems applying basic knowledge and techniques (see below)
at various scales of difficulty and complexity: simple calculations, modeling problems
where verbally expressed relationships must be put in algebraic form and deductions
made, and problems involving the interplay between graphical and algebraic
perspectives.

Approach problems multi-representationally, using algebraic, graphical, numerical, and
verbal perspectives, and effectively using technology when appropriate.


Present logical reasoning and proofs in various mathematical contexts (e.g. algebra,
geometry, modeling), and provide careful explanations of problem formulations,
solutions, and procedures when required.

Students should also acquire specific skills and knowledge of mathematics. The items below are
not intended to be exhaustive, but only to indicate the scope and level of proficiency implicit in
the Algebra II and Geometry expectations. The items do not include everything about the
mathematical sciences that would constitute an exemplary high-school program, such as basic
statistics, additional topics in geometry, and computing.

Students should be able to

Use correct conventional notation for arithmetic and algebraic expressions and formulas,
including parentheses, order of operations, equality and inequality signs.

Understand the real numbers as decimal expansions, represent them as points on a line,
graphically interpret order relations for real numbers, and perform order and absolute
value comparisons using arithmetic.

Perform, reliably by hand, all the basic operations of arithmetic, including division,
especially as applied to fractions. Convert between ordinary fractions, decimals and
percents, by hand in simple cases, and with calculators in more complicated ones. Use
calculators to supplement and leverage skill in hand calculation when appropriate.

Interpret and compute with integer and fractional exponents, radicals, and scientific
notation.

Compute and simplify sums, differences, and products of general polynomials.

Model linear relationships both algebraically in conventional forms, and graphically,
interpret coefficients in graphical terms, e.g. as slopes and intercepts, and solve problems
relating these representations to each other.

Solve quadratic equations by factoring, completing the square, and the quadratic formula,
and use these techniques to solve applied problems, and obtain a rough hand sketch of the
graph of a quadratic equation from its algebraic expression.

Solve equations that can be reduced to linear or quadratic equations by hand and use a
graphing calculator to estimate solutions of equations not feasibly solvable by hand.
Solve simple systems of equations.

Solve inequalities and use inequalities and absolute value to specify constraints on sets of
numbers.


Add, multiply, and divide rational expressions, that is algebraic fractions, and reduce
them to lowest terms, simplify compound fractions, and perform simple polynomial
division.

Understand the function concept and the use of function notation, know how functions
can be represented by graphs, draw inferences from the graphs about the function (values,
maximum values, increasing/decreasing, solutions of related equations, etc.).

Solve elementary counting problems involving permutations and combinations and apply
these to probability calculations.

Use the definition and properties of common and natural logarithms to solve simple
exponential equ