A few days ago I experienced one of the most inspiring moments as an algebra teacher… I gave my 9^{th} grade students a quiz on solving equations (one-step, two-step, multi-step, literal, etc.). One of the questions was a word problem that involved buying a season pass ticket to an amusement park versus buying single passes and making multiple visits to the park. The first part of the question asked students to determine how many trips to the park they would have to make in order for the season pass to be the better deal. The second part asked the students to write an equation to model the situation. The third part asked the students to solve the equation. Most of the students immediately solved the problem by writing and solving an equation. When they read the second and third parts of the problem they were confused because they had already completed both parts in the beginning. I was excited!!!!

This is why I was excited… The students were initially asked to solve the problem using any method (it was an open-ended question). Most of the students immediately wrote and solved an equation because that was their first thought. These students were ahead of the test question! They were already “thinking algebraically” before the question asked them to think algebraically.

After I collected the quizzes the students told me they were confused by the problem and wondered whether they answered it incorrectly. I told them they answered the question exactly the way they should have. I told them they were thinking algebraically and that is how they should be thinking. They were pleased with my response!

The goal of algebra teachers should be to help students think algebraically. When this happens, students begin to look at problems differently. They begin to generalize situations and find solutions quickly (and accurately). Thinking algebraically is a higher level of thinking that most students (and adults) never achieve. Most of my 9^{th} grade students are already thinking algebraically! As much as I would like to take full credit for this, I can’t. Their teachers before me did a phenomenal job and that makes my job easier.

I have been teaching college level precalculus for several years. A running theme of concern has been the lack of preparedness of my students for the course. The struggling students somehow place into the course, but clearly are not prepared. My assessment is that the students’ basic algebra skills are weak. But what happens when a student takes algebra for the first time, but are not prepared? Why are some students ready for algebra and some students struggle with the basic algebraic concepts covered in Pre-algebra or Algebra 1 courses? What do you do as a teacher when you are faced with the challenge of bridging the gap between arithmetic and algebra? How do you incorporate these concepts into your lessons without losing algebra “teaching time?”

This is an issue many Algebra 1 teachers face. The common concern is that students taking Algebra 1 lack basic arithmetic skills. But these skills are necessary for success in Algebra 1. For example, many students struggle with adding fractions. What happens when those same students have to solve equations with rational expressions? If they have not mastered adding fractions, they will not be able to solve equations with rational expressions or they will experience difficulty when faced with these problems.

To me the answer is clear… Teach students so that they master basic arithmetic skills before they enter Algebra 1. This charge is for elementary school teachers. Here is the reality… This is not always accomplished. Elementary school teachers probably have their reasons for why this is not happening, across the board. In the meantime, students are required to take Algebra 1 with whatever skills they have acquired. This presents a problem to secondary teachers who have students entering Algebra 1 lacking the basic skills needed to learn and master basic algebra concepts.

How do you bridge that gap as an Algebra 1 teacher? What does that bridge look like? How do you help these students without hindering the advancement of the students who were fortunate to have mastered these skills?

These are very valid questions with many valid answers. What are your thoughts? What have you done in this situation?

AKA: Order of Operations / Operator Precedence / PEMDAS / BEMDAS

Although there is so much you can do with math, without structure there is no point. If there were no rules to follow, it would be chaos – which is a fascinating math-logical reasoning in and of it-self. The process of the Order of Operations is to define the rules in solving equations. Used in basic arithmetic to calculus, this process is of the utmost importance in methods utilized in Mathematics because it sets the foundation for solving anything.

History

No one seems to be sure exactly when the Order of Operations took precedence in figuring out the… order of operations. The presence of this procedure can be considered to be influenced by the grammar and language when describing an equation (Peterson, 2000). On the website Dr. Math, it is mentioned that “multiplication has precedence over addition.” I take…

I recently had a conversation with a young lady who claims to “love” math. I was very excited for the young lady. Then she proclaimed “I do not see the connection between precalculus and calculus.” I wanted to ask her to explain herself, but I wasn’t sure I wanted to hear her explanation. I asked myself how someone could have such affection towards math, but not see the connection between the two subjects.

Precalculus is the study of functions, their graphs, and their properties. It builds the foundation for calculus. I spend several hours each week preparing my students for calculus and higher level math courses via precalculus. Some of my former students stop by my office to thank me for preparing them for calculus. If there is no connection, what am I doing with my life!!

I am saddened that this young lady feels this way about precalculus and calculus. I wanted to interrupt her and explain the connection right then and there, but time would not allow. The most unnerving part is that she aspires to teach math and claims to be “good at it.” What?!

I hope we one day cross paths again and I can revisit this conversation so I can get a better understanding of her proclamation that there is no connection between precalculus and calculus. This statement ignited within me concern for all the students who encounter teachers with this same attitude towards math.

What are your thoughts? How do you explain the connection between precalculus and calculus?

Calculus is a specified field of mathematics. Limits, derivatives, integrals, and infinite series are all applied to various types of functions (i.e., linear, quadratic, exponential, trigonometric, etc.). They each depend on the understanding of basic arithmetic as well as specific concepts of other branches of secondary level mathematics such as algebra, geometry, and trigonometry. Prior to taking a calculus course students should develop specific skills in each of these areas of study.

Below, I briefly discuss key skills and concepts required from three branches of mathematics (algebra, geometry, and trigonometry) to prepare students for calculus. I discuss this material as it relates to differentiation, a fundamental component of calculus.

Algebra (Functions)

Differentiation converts formulas into other formulas. These formulas are functions. Algebra is the study of functions. Since functions are the object of differentiation, it is imperative that students entering into a course of study in calculus have a thorough understanding of functions.

Many secondary students can follow instructions to substitute a value into an equation, but the study of functions will provide a greater understanding of why the method of substitution works and why a relationship exists between the parameters.

Geometry (Graphs)

The result of differentiation is a derivative. Derivatives are used to calculate the slope of a line tangent to a point on the graph of a function. A student can find an estimated value using the slope of a line secant to the same graph, but calculus makes it possible to find the exact value. Students must have a thorough understanding of graphs, slopes, and functions to understand the use of derivatives with respect to graphs of functions.

Trigonometry (Angles, Periods of revolution)

Trigonometry is the study of trigonometric functions. Trigonometric functions are used to calculate measures involving triangles, periods of revolution, etc. During the course of a calculus class students will encounter problems involving trigonometric functions which are useful for explaining periodicity. Students studying calculus should understand trigonometric functions, their properties, and their graphs.

As discussed above, students preparing to take calculus should have a specific set of skills relating to calculus content. The major subjects required for the development of these skills, and the understanding of the related concepts, are algebra, geometry, and trigonometry. More specifically, students need to have a thorough understanding of functions, graphs, slopes, and trigonometric functions, to name a few, in order to effectively prepare to enter into a calculus course.

Although students have the benefit of the use of technology to solve many problems, they must have an understanding of the relationship between the functions used, their graphs, and the derivative so they can understand the output generated from the use of technology. Simply plugging values into a technological tool does not provide this level of understanding. In most cases, students will not have access to technological tools in college level math courses (see “Calculators Not Allowed” on this blog site), so it’s best to learn and understand the concepts without these aids.

Do you have any other ideas about concepts students should learn prior to taking a calculus course?