Preparation for College Level Mathematics
Developmental Mathematics courses are used as preparation for a variety of math courses: ·College Algebra (and Pre-Calculus) ... the start of 'math intensive' courses ·Math for elementary education majors ·Liberal arts math (and related 'survey' courses) ·Technical mathematics ·Business mathematics

Preparation for College Algebra and Pre-Calculus The preparation needed for math intensive courses is guided by the needs of traditional college algebra & pre-calculus courses as well as the reform curriculum described by MAA's "CRAFTY" documents and AMATYC's "Right Stuff" project. Clearly, the preparation needs of students will depend on whether their college offers a traditional or a reformed college algebra program. Also, we recognize that some institutions offer college algebra as a general education course that is definitely not math intensive, while other colleges offer a college algebra course that is essentially pre-calculus. We conclude that the preparation needed for a non-math intensive college algebra course is similar to that needed for a reform college algebra course. Here we describe the range of needs involved with both types of programs (one traditional and math intensive, the other reform or general education); all developmental math programs should address all of these areas, with an emphasis on those areas that meet local needs. Our primary references are MAA materials -- CUPM 2004 for general needs, and CRAFTY for reform efforts.

Preparation for Elementary Education Math and Liberal Arts Math We treat the preparation needs of these courses in one group because of similarities. In some colleges, these are the same courses. In others, the courses are different but share content and emphasis. Our conclusion is that these courses depend more on creating 'mathematical maturity' in students and less on specific content. The mathematical needs for students going into these courses can be met by the content described "for academic success" (the third pillar), with a focus on quantitative literacy. Some additional information is provided in MAA CUPM 2004 materials, as seen below.

Preparation for Technical Mathematics and Business Mathematics
The label 'technical mathematics' covers a broad collection of courses -- some similar to developmental mathematics in content and sophistication, while others are closer to traditional calculus in nature. Business mathematics also refers to a range of courses. Because of these complexities, we do not identify any particular content in developmental mathematics to prepare students for either technical mathematics or business mathematics. We believe that a starting point to meet the needs of technical mathematics courses is the material listed for the second pillar -- preparing students for other courses (in technology).

Developmental Mathematics Content Needed for College Mathematics

Students who intend to continue beyond developmental mathematics (what about actually graduating?) to higher level mathematics courses need a deep understanding of elementary concepts/procedural skills and the analytical and problem solving skills to successfully complete work at the next level. They also need to recognize that mathematics is a 'foreign language' (but it isn't) (Yes it is! "MATHSPEAK") (using both specialized vocabulary and symbols [what about structure?]) and work towards fluency in that language in all developmental mathematics courses. Students need to realize that some knowlege in math (such as symbol use, the "language" part) simply needs to be memorized, while mathematical concepts should be UNDERSTOOD.

Students [should] address problems presented as real world situations by creating and intepreting mathematical models (I don't know what this means.) [there is so much concentration on 'real world' that this is losing meaning - real world to whom? Us or students?]. Solutions to the problems are formulated, validated, and analyzed using mental, paper and pencil, algebraic, and technology-based techniques as appropriate. [CRAFTY: College Algebra Guidelines]

Students' expertise should include all of the following:

Number Sense: Before students can make sense of higher (Is "higher" really college algebra or pre-calculus? If so, say college algebra.) mathematics, they need to have developed an adequate basic number sense. For example, if they do not understand that one-third is larger than one-eighth (or that -1/3 is less than -1/8) and why, they will have a hard time later seeing the function y = 1/x (x not zero) as a decreasing function. If they don't know the difference between 3 ^ 2, 3 x 2, (-3)^2, or -3 ^ 2, they will have difficulty with order of operations and exponential functions. Some specific outcomes include the ability to:
Order sets of numbers (rational and/or irrational; rationals including integers, decimal, fraction, and percent forms). Understand place value and the relative magnitudes of numbers (What about rounding, they are terrible at rounding.)
Develop intuitions about and the ability to communicate verbally and symbolically the effects of common operations on numbers (I don't know the intent of this sentence. What are "common" operations? Dose it mean arithmetic? Why "intuitions?" Why not just plain ability or skill?) (I think intuition is the right idea. Students need to start understanding what is going on behind the numbers, not just plug and chug.)
(How about: "Develop a depth of understanding of the arithmetic operations that will enable the student to communicate both verbally and symbolically the effects of the operations.")
Interpret operations concretely in multiple contexts and situations (e.g., subtraction is not always "take away" and multiplication is not always repeated addition). (Why not include proportional reasoning in this list?)(I think estimation is also a important skill to have, and i mean not just front-end rounding, but understanding how to manipulate the numbers to get a "down and dirty" guess....I guess I mean not to memorize how to do an estimation procedure but how to use it intuitively.)

Variable Sense: Understanding the concept of a variable is often taken for granted, but is critical for a student to make progress in algebraic thinking and problem solving. (How do we know this?) Students who believe that "x" is only an "unknown" to be found will have difficulty understanding the role of variables in functions. (Of course, just one more reason we should use a function approach.)
Some specific outcomes include the ability to:
Identify the nature of a variable from the context (place holder, unknown, property)
Apply appropriate replacement sets for variables, and use the concept of parameter (such as a,b,c in ax^2 + bx + c)
Not only perform procedures (such as simplifying exponential expressions and factoring polynomials), but communicate these processes clearly
Correctly use notation with variables (coefficients, exponents, subscripts) in simple and slightly complex expressions
Apply basic properties of real numbers to both simplify, evaluate, and expand expressions
Understand that an equation like x + x + x = 12 has only one solution, while x + y = 12 has infinitely many solutions(Move to equation sense, or delete.)

Equation Sense: Understanding basic principles about equality and what it means to "solve" an equation are critical to progress in mathematics. This extends to modeling first with one equation and then with systems of equations. Understand the difference between 'simplify an expression' and solve an equation.'

Some specific outcomes include the ability to: Know what a solution means in terms of an equation - in one var, in two vars...and correctly interpret the results in the context of the problem.
Use basic properties of equations (linear, quadratic, exponential, radical, rational) to solve equations. Logarithmic too??
Use numeric (or graphical) methods (graphing calculator and/or other technology) to solve any equation, and correctly interpret the results in the context of the problem.
Make reasonable choices between numeric [call it a table], graphic, and algebraic methods for solving types of (delete) equations
Use basic properties of inequalities to solve inequalities (linear, quadratic, absolute value.)
Model problems with multiple conditions with linear equations or inequalities with up to 3 variables
Use basic properties of equations, or technology to solve simple systems of equations in 2 or 3 variables.
Communicate accurately special cases of equations and systems of equations (involving contradictions and identities)

Function Sense: Understanding the concept of functions and rates of change, and illustrating that understanding effectively using multiple representations (symbolic, numeric, graphic, and verbal) is a basic skill. (Awkward wording and non-parallel value.) Starting with linear functions and progressing on to exponential, and to some of radical, rational, logarithmic, and/or polynomial functions is basic to understanding higher level mathematics. (How about "Understanding representation and behaviors of basic functions such as, linear, quadratic, exponential, radical, rational, logarithmic, and/or polynomial functions is crucial to understanding higher level mathematics.") Students should be able to fit an appropriate curve to a scatterplot and use the resulting function for prediction and analysis. [CRAFTY: Functions & Equations and Data Analysis] (How can we do this with no mention of statistical methods?)
Some specific outcomes include the ability to:
Identify the appropriate domain for any basic function
Calculate the rate of change at a point, and identify this as either the slope of the curve or the slope of the tangent line
Identity the input(s) and output for a function; communicate the meaning of an ordered pair
Identify the basic nature of a graph (given the graph, classify it as linear, exponential, radical, rational, or poynomial)
Change fluently between any two <omit "two"> representations of a function. (Missing are the behaviors that are significant: Increasing/decreasing, max/min, zeros, positive/negative, and range as well as domain and average rate of change. All of these are easily taught with a graphing calculator and a function approach.)

Geometric Sense: Especially for the elementary education majors, a basic understanding of geometry and measurement is important. An important piece should be a conceptual understanding of different dimensions, including the differences between the concepts of perimeter, area, and volume. Also important would be doing work with irregular shapes as well as basic shapes like circles, triangles, rectangles, cylinders, prisms, and spheres.
Some specific outcomes include the ability to:
Calculate perimeters, areas, volumes (in U.S. Standard or metric units) of basic and irregular shapes.
Change measurements and/or rates between the U.S. Standard and metric systems. (Why not Pythagorean relationships?)

Graphing Sense: The algebraic functions that are graphed in developmental mathematics courses all have some commonalities that students need to recognize. Each graph is a picture of the solutions of the equation. Often intercepts, slopes, and areas of increase/decrease are of interest, no matter what the function is. In real-data applications, interpreting what these concepts mean in the context of the particular problem situation is critical.
Some specific outcomes include the ability to:
Use technology for computing outputs, calculate sufficient ordered pairs to draw a graph of reasonable accuracy. (This is so 1950's. Pencil and paper graphing is a worthless task.) (I disagree, using only technology hampers deep understanding of the mathmatical concepts behind graphing.)
Given a function and domain, create a reasonable coordinate system showing the domain and range (Pencil and paper graphing is a worthless task.) (Disagree, see above)
Find intercepts for linear functions without the aid of technology (Why just linear functions?? With technology, all functions are accessible.)
Using technology, find the intercepts of any basic function; state when an intercept does not exist for a particular function
Find the slope for linear functions without the aid of technology (without the aid of technology IS NOT NEEDED.)
Using technology, find the slope at a point for exponential or polynomial function and interpret the meaning of this slope (Without calculus?)
Interpret the meaning of slope and intercepts in the context of authentic application problems.

Discrete: Students need a basic understanding of sets, Venn diagrams, counting methods, and probability. In addition, representing data using graphs, matrices, and measures of central tendency would help them prepare for a finite math or discrete math class. Some specific outcomes include the ability to:
Perform basis operations on sets (union, intersection), and use basic set notation
Use counting methods on small finite sets, including operations on those sets
Calculate probabilities involving selections from sets (with and without replacement, conditional)
Represent data in matrix form, including the use of a spreadsheet for this purpose; produce a reasonable graph (Why a spreadsheet? A graphing calculator is much more appropriate, and requires no time spent on learning new software.) (Teaching how to use spreadsheets is more real world that graphing calculators. Being able to produce different types of graphs is a skill that is often used in today's jobs.)

AMATYC's Right Stuff project (reference material)
Jack's note: Good material here at (Right Stuff); the problem -- it is a collection of materials from individuals, so it does not provide the same kind of authoritative source that MAA has (Crafty is a committee viewpoint, so what we see is at least an agreement among a group, however, small). I see Right Stuff as background reading materia, not primary source.

http://www.maa.org/t_and_l/urgent_call.html
Although this document specifically addresses College Algebra, I believe that the flavor of the document applies equally to the levels of algebra leading up to College Algebra as well. At least it's worth reading and thinking about, as we search for new directions for our 21st century developmental math program.
Jack's note: This also is a one-person persepctive; good background, not a primary source

Preparation for College Level Mathematics

Developmental Mathematics courses are used as preparation for a variety of math courses:

· College Algebra (and Pre-Calculus) ... the start of 'math intensive' courses

· Math for elementary education majors

· Liberal arts math (and related 'survey' courses)

· Technical mathematics

· Business mathematics

Preparation for College Algebra and Pre-CalculusThe preparation needed for math intensive courses is guided by the needs of traditional college algebra & pre-calculus courses as well as the reform curriculum described by MAA's "CRAFTY" documents and AMATYC's "Right Stuff" project. Clearly, the preparation needs of students will depend on whether their college offers a traditional or a reformed college algebra program. Also, we recognize that some institutions offer college algebra as a general education course that is definitely not math intensive, while other colleges offer a college algebra course that is essentially pre-calculus. We conclude that the preparation needed for a non-math intensive college algebra course is similar to that needed for a reform college algebra course. Here we describe the range of needs involved with both types of programs (one traditional and math intensive, the other reform or general education); all developmental math programs should address all of these areas, with an emphasis on those areas that meet local needs. Our primary references are MAA materials -- CUPM 2004 for general needs, and CRAFTY for reform efforts.

Preparation for Elementary Education Math and Liberal Arts MathWe treat the preparation needs of these courses in one group because of similarities. In some colleges, these are the same courses. In others, the courses are different but share content and emphasis. Our conclusion is that these courses depend more on creating 'mathematical maturity' in students and less on specific content. The mathematical needs for students going into these courses can be met by the content described "for academic success" (the third pillar), with a focus on quantitative literacy. Some additional information is provided in MAA CUPM 2004 materials, as seen below.

Preparation for Technical Mathematics and Business MathematicsThe label 'technical mathematics' covers a broad collection of courses -- some similar to developmental mathematics in content and sophistication, while others are closer to traditional calculus in nature. Business mathematics also refers to a range of courses. Because of these complexities, we do not identify any particular content in developmental mathematics to prepare students for either technical mathematics or business mathematics. We believe that a starting point to meet the needs of technical mathematics courses is the material listed for the second pillar -- preparing students for other courses (in technology).

Developmental Mathematics Content Needed for College MathematicsStudents who intend to continue beyond developmental mathematics (what about actually graduating?) to higher level mathematics courses need a deep understanding of elementary concepts/procedural skills and the analytical and problem solving skills to successfully complete work at the next level. They also need to recognize that mathematics is a 'foreign language' (but it isn't)

(Yes it is! "MATHSPEAK")(using both specialized vocabulary and symbols [what about structure?]) and work towards fluency in that language in all developmental mathematics courses. Students need to realize that some knowlege in math (such as symbol use, the "language" part) simply needs to be memorized, while mathematical concepts should be UNDERSTOOD.Students [should] address problems presented as real world situations by creating and intepreting mathematical models (I don't know what this means.) [there is so much concentration on 'real world' that this is losing meaning - real world to whom? Us or students?]. Solutions to the problems are formulated, validated, and analyzed using mental, paper and pencil, algebraic, and technology-based techniques as appropriate. [CRAFTY: College Algebra Guidelines]

Students' expertise should include all of the following:

: Before students can make sense of higher (Is "higher" really college algebra or pre-calculus? If so, say college algebra.) mathematics, they need to have developed an adequate basic number sense. For example, if they do not understand that one-third is larger than one-eighth (or that -1/3 is less than -1/8) and why, they will have a hard time later seeing the function y = 1/x (x not zero) as a decreasing function. If they don't know the difference between 3 ^ 2, 3 x 2, (-3)^2, or -3 ^ 2, they will have difficulty with order of operations and exponential functions.Number SenseSome specific outcomes include the ability to:

Order sets of numbers (rational and/or irrational; rationals including integers, decimal, fraction, and percent forms).

Understand place value and the relative magnitudes of numbers

(What about rounding, they are terrible at rounding.)Develop intuitions about and the ability to communicate verbally and symbolically the effects of common operations on numbers (I don't know the intent of this sentence. What are "common" operations? Dose it mean arithmetic? Why "intuitions?" Why not just plain ability or skill?)

(I think intuition is the right idea. Students need to start understanding what is going on behind the numbers, not just plug and chug.)(How about: "Develop a depth of understanding of the arithmetic operations that will enable the student to communicate both verbally and symbolically the effects of the operations.")

Interpret operations concretely in multiple contexts and situations (e.g., subtraction is not always "take away" and multiplication is not always repeated addition). (Why not include proportional reasoning in this list?)

(I think estimation is also a important skill to have, and i mean not just front-end rounding, but understanding how to manipulate the numbers to get a "down and dirty" guess....I guess I mean not to memorize how to do an estimation procedure but how to use it intuitively.): Understanding the concept of a variable is often taken for granted, but is critical for a student to make progress in algebraic thinking and problem solving. (How do we know this?) Students who believe that "x" is only an "unknown" to be found will have difficulty understanding the role of variables in functions. (Of course, just one more reason we should use a function approach.)Variable SenseSome specific outcomes include the ability to:

Identify the nature of a variable from the context (place holder, unknown, property)

Apply appropriate replacement sets for variables, and use the concept of parameter (such as a,b,c in ax^2 + bx + c)

Not only perform procedures (such as simplifying exponential expressions and factoring polynomials), but communicate these processes clearly

Correctly use notation with variables (coefficients, exponents, subscripts) in simple and slightly complex expressions

Apply basic properties of real numbers to both simplify, evaluate, and expand expressions

Understand that an equation like x + x + x = 12 has only one solution, while x + y = 12 has infinitely many solutions(Move to equation sense, or delete.)

: Understanding basic principles about equality and what it means to "solve" an equation are critical to progress in mathematics. This extends to modeling first with one equation and then with systems of equations.Equation SenseUnderstand the difference between 'simplify an expression' and solve an equation.'

Some specific outcomes include the ability to:

Know what a solution means in terms of an equation - in one var, in two vars...and correctly interpret the results in the context of the problem.

Use basic properties of equations (linear, quadratic, exponential, radical, rational) to solve equations. Logarithmic too??

Use numeric (or graphical) methods (graphing calculator and/or other technology) to solve any equation, and correctly interpret the results in the context of the problem.

Make reasonable choices between numeric [call it a table], graphic, and algebraic methods for solving types of (delete) equations

Use basic properties of inequalities to solve inequalities (linear, quadratic, absolute value.)

Model problems with multiple conditions with linear equations or inequalities with up to 3 variables

Use basic properties of equations, or technology to solve simple systems of equations in 2 or 3 variables.

Communicate accurately special cases of equations and systems of equations (involving contradictions and identities)

: Understanding the concept of functions and rates of change, and illustrating that understanding effectively using multiple representations (symbolic, numeric, graphic, and verbal) is a basic skill. (Awkward wording and non-parallel value.) Starting with linear functions and progressing on to exponential, and to some of radical, rational, logarithmic, and/or polynomial functions is basic to understanding higher level mathematics. (How about "Understanding representation and behaviors of basic functions such as, linear, quadratic, exponential, radical, rational, logarithmic, and/or polynomial functions is crucial to understanding higher level mathematics.") Students should be able to fit an appropriate curve to a scatterplot and use the resulting function for prediction and analysis. [CRAFTY: Functions & Equations and Data Analysis] (How can we do this with no mention of statistical methods?)Function SenseSome specific outcomes include the ability to:

Identify the appropriate domain for any basic function

Calculate the rate of change at a point, and identify this as either the slope of the curve or the slope of the tangent line

Identity the input(s) and output for a function; communicate the meaning of an ordered pair

Identify the basic nature of a graph (given the graph, classify it as linear, exponential, radical, rational, or poynomial)

Change fluently between any two <omit "two"> representations of a function. (Missing are the behaviors that are significant: Increasing/decreasing, max/min, zeros, positive/negative, and range as well as domain and average rate of change. All of these are easily taught with a graphing calculator and a function approach.)

: Especially for the elementary education majors, a basic understanding of geometry and measurement is important. An important piece should be a conceptual understanding of different dimensions, including the differences between the concepts of perimeter, area, and volume. Also important would be doing work with irregular shapes as well as basic shapes like circles, triangles, rectangles, cylinders, prisms, and spheres.Geometric SenseSome specific outcomes include the ability to:

Calculate perimeters, areas, volumes (in U.S. Standard or metric units) of basic and irregular shapes.

Change measurements and/or rates between the U.S. Standard and metric systems.

(Why not Pythagorean relationships?)

: The algebraic functions that are graphed in developmental mathematics courses all have some commonalities that students need to recognize. Each graph is a picture of the solutions of the equation. Often intercepts, slopes, and areas of increase/decrease are of interest, no matter what the function is. In real-data applications, interpreting what these concepts mean in the context of the particular problem situation is critical.Graphing SenseSome specific outcomes include the ability to:

Use technology for computing outputs, calculate sufficient ordered pairs to draw a graph of reasonable accuracy. (This is so 1950's. Pencil and paper graphing is a worthless task.)

(I disagree, using only technology hampers deep understanding of the mathmatical concepts behind graphing.)Given a function and domain, create a reasonable coordinate system showing the domain and range (Pencil and paper graphing is a worthless task.)

(Disagree, see above)Find intercepts for linear functions without the aid of technology (Why just linear functions?? With technology, all functions are accessible.)

Using technology, find the intercepts of any basic function; state when an intercept does not exist for a particular function

Find the slope for linear functions without the aid of technology (without the aid of technology IS NOT NEEDED.)

Using technology, find the slope at a point for exponential or polynomial function and interpret the meaning of this slope (Without calculus?)

Interpret the meaning of slope and intercepts in the context of authentic application problems.

: Students need a basic understanding of sets, Venn diagrams, counting methods, and probability. In addition, representing data using graphs, matrices, and measures of central tendency would help them prepare for a finite math or discrete math class.DiscreteSome specific outcomes include the ability to:

Perform basis operations on sets (union, intersection), and use basic set notation

Use counting methods on small finite sets, including operations on those sets

Calculate probabilities involving selections from sets (with and without replacement, conditional)

Represent data in matrix form, including the use of a spreadsheet for this purpose; produce a reasonable graph (Why a spreadsheet? A graphing calculator is much more appropriate, and requires no time spent on learning new software.)

(Teaching how to use spreadsheets is more real world that graphing calculators. Being able to produce different types of graphs is a skill that is often used in today's jobs.)References:

MAA CRAFTY College Algebra Guidelines (2007)

AMATYC's Right Stuff project (reference material)

Jack's note: Good material here at (Right Stuff); the problem -- it is a collection of materials from individuals, so it does not provide the same kind of authoritative source that MAA has (Crafty is a committee viewpoint, so what we see is at least an agreement among a group, however, small). I see Right Stuff as background reading materia, not primary source.

http://www.maa.org/t_and_l/urgent_call.html

Although this document specifically addresses College Algebra, I believe that the flavor of the document applies equally to the levels of algebra leading up to College Algebra as well. At least it's worth reading and thinking about, as we search for new directions for our 21st century developmental math program.

Jack's note: This also is a one-person persepctive; good background, not a primary source

MAA Curriculum Guide 2004