Pre-Algebra: The Foundation of the Algebra Strand Through Calculus
Within the K-12 scope and sequence, pre-algebra marks the beginning of the second major developmental milestone spanning kindergarten through calculus. The first major development, typically within the domain of grades K- 5 math instruction, is the concept-procedure integration of the four operations through fractions and decimals and the 400 math facts.
The algebra strand relies heavily on the developments achieved in the elementary strand because the major fundamental for elementary grades mathematics is addition, subtraction, multiplication, and division through fractions and decimals when we know how much we have; and the major fundamental for algebra is also addition, subtraction, multiplication, and division through fractions and decimals, but in algebra, we may not know how much we have, because in algebra, there are variable expressions and variable equations. Additionally, the combination of learning fractions and decimals across the four operations and developing automaticity with the 400 math facts during the elementary grades directly supports the development of automaticity with the integer facts and rational numbers applications across the four operations necessary for simplifying variable expressions and solving variable equations.
Getting Ready for Pre-Algebra
Completing fifth grade does not necessarily indicate students’ readiness for pre-algebra – only the full concept-procedure integration of the four operations through fractions and decimals and automaticity with the 400 math facts does. Therefore, readiness for pre-algebra can occur at any time the student(s) has achieved this developmental integration, because if students are required to begin pre-algebra without having achieved the necessary first major fundamental development, the prognosis for success in algebra is questionable.
To cover this voluminous and essential foundation in elementary grades mathematics, I have created two seminars: Making Math Real: The 4 Operations and the 400 Math Facts (4 Ops) and Making Math Real: Fractions, Decimals, and Advanced Place Value (FDAP). At 12 days each, these are the two longest seminars the Making Math Real Institute offers, and since they provide the critical foundation for the entire algebra strand, they are the only two large-scale (seminars of 10 or more days) institute courses offered once per year, every year.
The Working Memory Demands for Algebra are Different Than Those for Elementary Mathematics
Elementary math provides the initial fundamentals of all future math, therefore its developmental/perceptual demands are basic. In elementary math the developmental/perceptual focus for students is to activate their working memory/comprehension picture entirely from the symbols, which is achieved by guiding the students from the concrete to the abstract. This means students’ original working memory/comprehension picture of all the math content is derived entirely from a direct concrete experience of the math, and once established, teachers structure the systematic transfer of the concrete working memory picture to the identical working memory picture expressed by the symbols: math symbols are a short hand code to express what is real.
Symbol imaging for numbers is therefore one of the principle sensory-cognitive developments that supports the working memory demands of elementary mathematics. Symbol imaging is the combination of the following four self-regulatory executive processes: perceive, hold, store, and retrieve sequences of numerical symbols. Symbol imaging is directly engaged both for learning the math facts and for concept-procedure integration of the four operations through fractions and decimals. In learning the math facts, symbol imaging allows us to fluently perceive, hold store and retrieve the sequences of symbols comprising each of the math facts; and in learning the four operations through fractions and decimals, symbol imaging is also required for successfully activating the concrete working memory picture from the symbols.
Once the algebra strand begins with pre-algebra, the working memory demands of the math change and become more sophisticated. In algebra, students will no longer activate working memory as a concrete experience expressed by the symbols, and this is why concrete models of algebra are strongly contraindicated. Instead of activating a concrete picture of the math from the symbols, in algebra, the perception of the interrelationship of the symbolic details activates the working memory/comprehension picture. This symbol-based detail relationship determines our comprehension of the algebra. For example, the detail interrelationship of the symbols indicates:
- The slope of a line and its y-intercept from an equation in slope-intercept form
- If an expression is in simplest form
- The difference between a perfect square trinomial from x(2) + bx + c when both are presented in product form
The sensory-cognitive development necessary for activating working memory from the interrelationship of the symbolic details is called detail analysis. Detail analysis is a higher order executive process than symbol imaging and depends on the development of symbol imaging as its basis, because to be able to analyze the interrelationships of symbolic details, students must be able to perceive, hold, store, and retrieve sequences of numerical symbols to support analyzing them mentally. Detail analysis refers to the ability to select out irrelevant details, focus in on key details and link key details back to the big picture of comprehension. Detail analysis also includes task monitoring as part of its function as a cognitive editing tool, enabling us to check and monitor our work as we proceed through problem solving.
All math has a developmental focus and a content focus. They are entirely distinct, yet the two interconnect constantly in support of one another. The developmental focus of pre-algebra is to engage and strengthen students’ applications of detail analysis in support of their activating and sustaining working memory by perceiving the interrelationship of the symbolic details. The mathematical content focus of pre-algebra is to introduce students to a new kind of problem solving: simplifying variable expressions and solving variable equations.
The Two Posters on Permanent Display In My Algebra Classrooms
In support of the new sensory-cognitive demands for activating working memory in algebra, I prominently place two posters in my classroom to which I refer consistently:
- Poster #1: Manage the Details, Your Grade Depends on it.
- Poster #2: Get Your Picture Before Expressing Any Transformation or Solution
The purpose of these posters is to help students become increasingly aware of their own working memory/comprehension pictures through the activation of detail analysis. The way to manage the details is by getting the picture (activating working memory) before expressing any next transformation or solution. The high incidence of detail errors in students’ algebraic problem solving (incorrectly referred to as careless mistakes) indicates detail analysis is not (fully) engaged, meaning working memory for algebra is not established, and therefore, students are proceeding through problem solving while not being able to successfully perceive the details. This is why I require all students to show all detail transformations and do not allow for mental math to replace it.
“I can do this math in my head”, boasts a student, hoping I will be impressed with the mental acumen on display. The expression and tone of my response do not indicate that I am impressed as I point to poster #1: “Doing the math in your head is for elementary school. We are in a new math universe in which the main purpose is to manage the details, so you are required to show you are managing the details by expressing every detail transformation.” It is not about the arithmetic of algebra,” I continue, “It is about your ability to manage all of the details throughout problem solving, no matter how easy you feel any particular detail is. This is helping us to learn to keep our picture for longer and longer periods without making detail mistakes. Managing the details is how we will continue to get a paycheck as adults.”
“But negative seven is close to positive seven,” laments another student who received no credit for a solution by not including the negative signage, feeling unfairly treated because a negative sign is such a small, little mark, rather than comprehending the details were not sufficiently well managed, and the difference in solutions is significant.
To What Does the “Pre” in Pre-Algebra Refer?
Over the last ten years I have observed a lack of consensus on what math constitutes PRE-algebra. The range of content presented in textbooks and in public/private school classrooms around the country is so varied, scattered, and disconnected, I cannot determine any common basis for what is intended by pre-algebra. The frequency with which content from algebra 1, algebra 2, and high school geometry is randomly dropped in pre-algebra students’ laps can create inappropriate scopes and sequences that may seriously impair their ability to learn the pre-algebra content. Some representative examples in which pre-algebra students are expected to:
- Solve problems with parabolas, but have yet to be introduced to the x-y Cartesian plane or the graphs of lines
- Solve exponential functions and they have yet to be introduced to constants, variables, expressions, or equations
- Solve surface area and volume of polyhedrons, 3-d space figures, and composite figures and have yet to be introduced to what prisms, pyramids, cylinders, cones and spheres are as well as finding the area of polygons.
I believe we can all agree on the meaning of the prefix, “pre-”. In the case of “pre-” algebra, there are distinct parameters that distinguish pre-algebra from algebra 1. The function of pre-algebra is to provide the initial foundation and problem solving tools from which all future algebraic processing will emanate. This is especially critical because the developmental demands of activating working memory for algebra are different from and more challenging than those for elementary mathematics, and students’ introduction to algebra needs to be successful and strong to support the higher level expansions coming in algebra 1, algebra 2, pre-calculus, and calculus.
Some Parameters That Distinguish Pre-Algebra From Algebra 1
The “pre-” in pre-algebra is about preparing students for most of what comprises algebraic problem solving: simplifying variable expressions, solving variable equations and graphing. Therefore, it is within the domain of pre-algebra that students learn the integer and rational numbers facts across the four operations, as these number facts will be applied throughout simplifying and solving for all algebra. The extent of simplifying expressions in pre-algebra is basic only, mostly to learn what constants, variables, expressions and equations are and the Order of Operations, GEMDAS, and not for combining like terms which, is an early algebra 1 unit in which students learn what a polynomial is, the degree of the polynomial, and expressing polynomials in standard form.
For solving equations, pre-algebra covers only the first four levels of solving equations in one variable and the first four levels of solving inequalities in one variable. Levels five through fourteen of solving equations in one variable are all covered in algebra 1. The parameter that distinguishes the pre-algebra equation solving levels from the algebra 1 levels is all pre-algebra level equations are ready to solve and all the algebra 1 levels require extra transformations and/or simplifying of expressions before the equations are ready to solve. The pre-algebra equation solving levels provide students with a solid foundation for all future equation solving: an equation is ready to solve when one variable expression in simplest form equals one constant expression in simplest form; and to be appropriate equations for pre-algebra, all variable expressions and all constant expressions are already in simplest form. Any variable or constant expressions requiring combining like terms, use of the distributive property, or having variable terms in both expressions are all exclusively parts of the algebra 1 equation solving levels.
In the ratio, proportion, percent unit, pre-algebra covers the four levels of fraction, decimal, and percent equivalence, the introduction of ratios, rates and proportions, the three forms of percent problem solving, and solving basic proportions with a single variable and three constants. Solving percent change and proportions with binomial numerators and/or denominators is covered in algebra 1.
The pre-algebra probability unit covers all basic probability including theoretical and experimental probability, simple, multiple, and compound events, dependent and independent events, dependent and independent events with and without replacement, and the counting principle. Permutations (nPr) and combinations (nCr) are typically presented in second semester algebra 2.
The pre-algebra linear graphing unit starts with graphing on the x-axis and solving levels one through four of inequalities. Pre-algebra follows with graphing on the x-y Cartesian plane, by first introducing solving equations in two variables and connecting the infinite solutions generated by 2 variables in one equation to the graphical expression of the infinite solutions as infinite points comprising a line. From this experience students learn slope-intercept and standard forms of linear equations. The extent of the linear graphing unit for pre-algebra is students’ ability to graph any line with positive, negative, and special slopes from any given equation, and to generate the equation of any graphed line with positive, negative, and special slopes. Algebra 1 continues this development by teaching students to generate the equations of lines algebraically without seeing the line on a graph.