Section 1.1 How We Write Affects How We Think
Learning Objectives
Present a sequence of algebraic manipulations with the equal signs lined up.
State an algebraic manipulation in words and then execute that manipulation correctly.
Welcome to college level mathematics! It is a well-known fact that many students struggle with college mathematics. There are many people who proudly announce that they took a college math course three or four times before they passed.
But what is less known are the reasons why students struggle with college level mathematics. Some will cite math anxiety, others may blame the instructor, and others will simply say that they're not math people. And there's probably a hint of truth to all of those.
A more basic reason for struggling with mathematics is that most students simply have not learned to think mathematically before they get to college. Maybe they've memorized some formulas and some basic ways to manipulate equations, but there's often a gap of understanding and the inability to communicate mathematics effectively.
The easiest way to spot this is when a student says, "I know what I'm doing, but I can't explain it." This might be good enough at lower levels of mathematics, where the goal is simply to get the answer, but college level mathematics is different. We want students to be able to explain what they're doing and why.
These materials are designed to help students bridge that gap so that they can become mathematical thinkers. But what is mathematical thinking? While it's hard to describe it in complete detail, it includes (and is not limited to) the following:
The ability to think logically and analytically to solve problems
The proper manipulation of symbols in the process of solving problems
The ability to communicate and explain the ideas behind algebraic manipulations and the reasons for using them
As you work your way through these materials, you should be finding yourself being able to do these things better through thoughtful practice and repetition. Just as with a foreign language, it's not enough to just say a phrase once to learn it. You need to do it over and over again, and you need to see it or hear in multiple contexts.
Activity 1.1.1. The Basic Presentation Expectations.
We're going to start with some basic ideas for communicating mathematics. For the purposes of this book, a complete presentation of a mathematical manipulation includes the following pieces:
A series of equations or expressions with the equal signs lined up vertically
An explanation of the manipulations on the right side of each manipulation
Here is an example of solving the equation \(3x + 4 = 16\) using a complete presentation:
Try it!
Using Activity 1.1.1 as a model, solve the equation \(5x - 7 = 23\text{.}\)
Solution.Activity 1.1.2. Lining up Equal Signs Vertically.
Lining up the equal signs is mostly a matter of organization and readability. Some students learned to do this because then they can use up the entire width of their paper and can cram in more problems per page. Unfortunately, as problems become more complex, this leads to all sorts of small errors that could easily be avoided by a more organized presentation.
Consider the following set of equations:
There are two errors embedded into these calculations. Notice how much your eyes have to go back and forth to match up the terms. Lining up the equal signs vertically reduce that distance and make it easier to see when terms change or disappear. It also helps you to keep track of what is on the left side of the equation and what is on the right side of the equation.
Try it!
Present the calculation in Activity 1.1.2 using a complete presentation. Be sure to fix the errors.
Solution.The two errors:
Line 2: Wrong sign on the 16
Line 3: Calculation error
Activity 1.1.3. A Common Presentation Problem.
There is a common way of writing equations that some people use which is less than ideal:
There are a number of problems with this. To start, there's not really an organized way to read this. The best way to think of it is that it breaks apart into four pieces with three of them overlapping each other. Another problem is that the reader is expected to simply know what's happening. This is sometimes at the root of students' complaints about "skipped steps."
Try it!
Write the calculation in Activity 1.1.3 using a complete presentation.
Solution.Activity 1.1.4. Describing the Steps.
One other aspect of taking the time to carefully write what you're doing is that it causes you to think more clearly about what you're doing. Consider the following equation: \(2x = \frac{1}{2}\text{.}\) A fair number of students see this and think they should "cancel out" the 2 from both sides. This is not correct. There are times you can cancel out numbers if one of them is in the numerator and the other is in the denominator, but this isn't one of them.
The act of explicitly stating what mathematical operation you're performing helps your brain to make categories of information. A more simple "error" of this type is that some students refer to all algebraic manipulations as "moving the term to the other side." Here are two examples of what that could mean to students:
Each of these is a different mathematical operation and a different algebraic step. It is not uncommon to see the following mistakes:
Usually, when I ask students to state in words what they did, they're able to see their mistake for themselves. So the act of stating the mathematical operation in words allows students to avoid errors.
Try it!
Solve the equations \(3x + 5 = 14\) and \(3x - 7 = 14\) using a complete presentation.
Solution.You will not always need to use a complete presentation. There are many times when it is tedious and unnecessary to state what is happening at every single algebraic step. This becomes more and more true as your basic algebra gets stronger and stronger. What ends up happening is that the steps that need to be described become less about the small details and more about the big picture. When solving for \(x\text{,}\) it may be enough to say, "Apply the quadratic formula" and not have to explain all of the individual arithmetic steps.
That being said, it is very important to develop the habit of keeping your equal signs lined up when writing math. If there is any one writing habit you develop from these worksheets, it should be that one. This one change by itself does a lot of things for you that you may not be consciously aware of.