“Math is not about numbers.”
The first time my prealgebra students hear me say this, they stare at me with confusion in their eyes. I’m sure some are silently thinking “OMG, my parents signed me up to take a class with a crazy woman. Of course, everyone knows that language arts is about words and math is about numbers.”
Math is the science of patterns. Mathematicians study patterns and relationships. They use a system of structured logic to analyze these relationships which are then often described using one or more of the languages of mathematics. At its core, math is pure logic.
Most people think math is the study of numbers because elementary school math primarily focuses on a branch of math called arithmetic. While elementary students do get glimpses of a few other branches of math such as algebra and geometry, most of their math studies involve adding, subtracting, multiplying and dividing different types of real numbers.
Algebra is the branch of math focusing on equations with variables. Each equation is a complete sentence that can be directly translated into a colloquially spoken language such as English, Spanish or Hindi. The beauty of a sentence that is translated into the language of algebra is that it can then be transformed using clearly defined properties and identities, such as the Order of Operations or the Identity, Associative, and Commutative Properties of Addition and Multiplication. These transformed equations can answer questions and reveal new insights into the original statement.
Educators use the term “Algebra” in two ways that differ from the definition of “Algebra” as a branch of mathematics. Algebra is the name of the first high school-level math course. It is also included as the name of a Common Core domain. These different definitions of the same word often cause confusion.
Common Core – Operations & Algebraic Thinking
The Common Core is a set of learning objectives for students in each grade. It is essentially a checklist of what students should learn in each grade. Each item on this checklist is called a standard. The standards are grouped by topics called domains.
One of the domains for students in grades K-5 is called “Operations & Algebraic Thinking.” The standards included in this domain are as follows:
Kindergarten
- Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
Grade 1
- Represent and solve problems involving addition and subtraction
- Understand and apply properties of operations and relationship between addition and subtraction.
- Add and subtract within 20.
- Work with addition and subtraction equations.
Grade 2
- Represent and solve problems involving addition and subtraction.
- Add and subtract within 20.
- Work with groups of objects to gain foundations for multiplication.
Grade 3
- Represent and solve problems involving multiplication and division.
- Understand properties of multiplication and the relationship between multiplication and division.
- Multiply and divide within 100.
- Solve problems involving the four operations, and identify and explain patterns in arithmetic.
Grade 4
- Use the four operations with whole numbers to solve problems.
- Gain familiarity with factors and multiples.
- Generate and analyze patterns.
Grade 5
- Write and interpret numerical expressions.
- Analyze patterns and relationship.
When elementary school students are working on any of the above topics included in “Operations and Algebraic Thinking,” they will often report that they are learning algebra. Elementary school teachers often shorten the name of this domain to simply “algebra.” Enrichment math classes for elementary students may even expand on the standards to include the introduction of variables.
While these concepts are essential for preparing students to eventually take algebra, mastering these learning objectives is not equivalent to taking a high school level algebra course. Most students who successfully master all of these concepts will need two additional years of math studies to be prepared for high school algebra.
High School Algebra
High school algebra is a full year course focusing primarily on algebra but normally also includes topics from geometry and statistics. While the topics covered in high school algebra have not changed significantly over the past several decades, the methods used to teach high school algebra have changed dramatically since the passage of No Child Left Behind (NCLB) in 2002, the adoption of the Common Core in 2010, and the passage of Every Child Succeeds Act (ESSA) in 2015.
Students still learn to solve and graph linear equations. They still perform operations with polynomials. They learn to factor, solve quadratic equations, and work with rational expressions and equations. A side-by-side comparison of the table of contents of a modern algebra book and one that was published in the 20th century would not reveal any dramatic changes. However, a side-by-side comparison of the content exposes dramatic changes.
Current algebra textbooks begin chapters with examples of how to solve specific types of problems. They are followed by short exercise sets students can complete by following the step-by-step examples. Most units conclude with a short test prep section.
These textbooks focus on HOW to use step-by-step procedures to complete problem sets. They provide no explanation of WHY certain techniques are used. Topics are presented as discrete modules rather than combined to provide increasing complexity. Training in logic is completely missing. Logic is the life giving blood of all mathematics. Logic is the unifying essence determining if a topic or process is a member of the set of mathematical systems.
Current algebra courses typically focus on how to solve specific problems likely to appear on standardized tests rather than on how to apply general problem-solving techniques that will train students to solve real-world problems outside the classroom. The most frequent questions I ask my algebra students are, “Why? Why can you do that step? Why is this problem difficult?”
During a recent class meeting, a student requested help by stating, “I don’t know what to do with this problem because it has a fraction as a coefficient.” Rather than provide the student with a step-by-step procedure to handle a quadratic equation with rational coefficients, I simply said, “Great! You identified your problem.” The student then almost immediately responded, “Oh, yeah, I can just factor out the fraction.” What I heard in that statement wasn’t just that the student could solve a particular problem, but he was able to use a problem solving strategy that could be applied not just to math but to life: Identify the problem by asking, “Why am I stuck?” (In this case the student was stuck because there was a rational coefficient.) Ask yourself, “How can I get rid of this problem?” (In this case, “How can I get rid of the rational coefficient?”)
A rigorous algebra course trains students to think logically and to break down complex problems into discrete manageable steps. I do not allow my students to use ‘shortcut’ formulas that they cannot prove. I challenge my students to derive the quadratic equation by completing the square of a problem where all the coefficients are replaced by variables. Less than half of the class expressed confidence in their ability to achieve this feat. Several raised their hands multiple times to express their lack of confidence. Each time I responded by asking, “Do you think you could figure out just one more step toward isolating x?” A few that needed this encouragement for almost every line. In the end, everyone was able to arrive at the quadratic equation. They were amazed at their own ability to combine multiple algebraic properties to solve a problem unlike anything they had ever attempted before. They demonstrated that they had achieved my year-long learning objective: To be able to use algebraic properties and deductive reasoning to solve problems.
Preparing for Algebra
It is essential that students are properly prepared before enrolling in algebra. Elementary students must first master operations with real numbers including: adding, subtracting, multiplying, and dividing whole numbers, integers, fractions, decimals, and percents plus order of operations, exponents and square roots. A student who demonstrates proficiency in ALL of these topics is ready to more closely examine WHY these arithmetic procedures work and apply the same procedures to expressions containing variables. This bridge between arithmetic and algebra is prealgebra. Students must master prealgebra before enrolling in algebra.
When do students take Algebra?
The average student in the United States takes algebra I in 9th grade. To prepare for algebra, students take middle school math that is equivalent to prealgebra.
Many school districts offer an accelerated math option for their top math students. These students often take algebra in 8th grade. Some school districts offer a double accelerated math option that allows a few select students to take algebra in 7th grade. Schools rarely allow students to take algebra before 7th grade.
Being allowed to register for algebra is not necessarily the same as being mathematically ready to take algebra. Schools often offer a less strenuous version of algebra for high school students who are not adequately prepared. I believe that course placement should be based on mathematical readiness and maturity rather than age or grade in school.
Students are ready for prealgebra if they can consistently, rapidly, and correctly do all of the following:
- Add, subtract, multiply, and divide whole numbers.
- Add, subtract, multiply, and divide integers.
- Add, subtract, multiply, and divide fractions and mixed numbers, including finding the lowest common denominator and reducing fractions to their simplest forms.
- Add, subtract, multiply, and divide decimals.
- Solve problems involving percents.
- Convert between fractions, mixed numbers, decimals and percents.
- Raise whole numbers to exponents.
- Take the square root of perfect squares.
- Use the Order of Operations.
- Solve simple word problems.
- Demonstrate that they are mature enough to handle a serious math classroom environment, homework requirements, and textbook.
I frequently meet parents who want to push their children into prealgebra or algebra before they are ready. It is essential that students do not enroll in advanced math classes until they demonstrate that they have mastered all the content of prerequisite courses. Students who are advanced before they are ready will end up with ‘holes’ in their math education that will limit their long-term math development.
Students are ready for algebra if they can:
- Apply all prealgebra arithmetic requirements to expressions containing variables.
- Simplify and work with radicals.
- Understand and apply rules of exponents.
- Apply basic mathematical properties and definitions to transform algebraic expressions.
- Apply properties of equality to solve simple equations.
- Translate phrases and sentences into algebraic expressions and equations.
Recommendation to Parents
Each child is a unique individual who will develop at a unique pace. Don’t push your child to advance to a class before being adequately prepared or allow your child to be trapped in a class with little or no new content to learn. Talk to your child’s teacher in the spring about potential class placement for the following academic year. Do not wait until summer or after school starts. If you disagree with the proposed class placement, ask for an objective test of skills. Use the test results as a diagnostic tool to identify areas of weakness.
Parents of elementary school students should inquire about advanced math tracks. Although algebra is rarely offered to students in public schools before 7th grade, students must often begin preparing for accelerated math in 4th or 5th grade in order to eventually qualify for 7th or 8th grade algebra.
Be aware that students who do not take algebra in the earliest year offered by their school district rarely have the ability to later move up into this highest math track. Early enrollment in algebra not only allows students to take more advanced math in high school, but also allows them to be prepared to take advanced high school courses in physics, chemistry, biology, computer science, and economics. Equally important, it ensures that they will take classes with the most serious and studious students in the school district which greatly affects their choice of friends.
Parents who wish for special accommodations to allow their child to take algebra earlier than it is offered through their school district, should consult someone with experience working with children who are radically accelerated to insure that they consider not only the academic implications but also the social and emotional aspects of the proposed placement.
As parents, we need to open the doors to learning but we must not push a child through those doors without adequate preparation. Carl Gauss, who is sometimes referred to as The Prince of Mathematicians, once said,
“It is not knowledge, but the act of learning, not the possession but the act of getting there, which grants the greatest enjoyment.”
Properly presented to prepared students, Algebra introduces them to this greatest enjoyment.