If a student has trouble solving a basic equation in 7th grade, they will also struggle to solve quadratic, exponential, or logarithmic equations. In these cases, it’s important to step in early to overcome these difficulties and keep the student’s relationship with algebraic reasoning from becoming more complicated.
In this article, we discuss the most common mistakes Secondary students make when solving math equations, how to identify these difficulties, and the best strategies to help students overcome them. Let’s keep them from developing math anxiety and make learning an enriching experience!
Why is algebra important?
When we talk about algebra, equations come to mind, but it goes further: algebraic reasoning involves generalizing and writing expressions, equations, and functions using algebraic language and symbols to achieve precision.
Algebra is used to solve problems, create mathematical models, and address situations from everyday life and from different fields of human knowledge.
If our students struggle to solve simple equations when we introduce linear equations in Secondary, they will also have trouble solving quadratic, exponential, or logarithmic equations in the future.
First, it’s important to watch for recurring mistakes. Second, we need to understand what lies behind the mistakes so we can give students specific, targeted feedback, always with the goal of helping them overcome them. It’s not enough to spend a week reviewing terms; we need to understand exactly what the procedural mistakes are. These difficulties will keep coming up throughout the school year and can become an insurmountable obstacle in learning math. Let’s see how to identify them!
How can we identify our students’ difficulties and mistakes?
At the start of the school year, prevention is better than cure. At this point, it’s essential to observe students’ work for a while to assess their knowledge of solving equations and identify any problems:
- We assess whether they know certain key terms, such as equality and equivalence. Understanding what an equation means is essential, since it helps clear up many doubts when students get stuck or are solving problems.
- We identify the mistakes they make and analyze them in depth, not just by correcting the answers, but by working alongside them as they solve problems to understand where they’re struggling.
In the article My Students Struggle to Solve Basic Equations, Michelle Russell shares a strategy for identifying the most common difficulties and mistakes in the math classroom. It involves asking students to individually make up an equation and solve it. Then they have to write down the steps they followed and explain their reasoning out loud. As teachers, this exercise is very valuable because it lets us analyze their answers and classify common mistakes.
3 common mistakes in solving equations
You’ve probably come across some of these difficulties in the math classroom when students solve equations:
1. Understanding the equal sign (=) in equations. Sometimes, if there’s a 0 on one side of the equation when subtracting, students may ignore or even drop the equal sign. If this all-important sign is missing from the solving process, it points to a difficulty understanding the concept of equivalence, one of the key terms we mentioned earlier.
2. Fractions in equations. In some cases, students can’t solve the problem because they come across a fraction in the equation and don’t know how to deal with it. Problems involving fractions are both arithmetic and algebraic, and they may point to a difficulty understanding division of fractions, its relationship with multiplication as the inverse operation, and the concept of a reciprocal. It could also be an algebraic mistake: not recognizing 2/3x as 2/3 times x, the student subtracts 2/3 from both sides of the equation.
3. Combining unlike terms. Why does the student keep combining different terms? Even though we constantly model the correct way to combine like terms, sometimes students don’t know how to proceed and simply guess. This may be due to the student’s anxiety about solving the problem, an operational understanding of the equal sign, and/or a lack of understanding of what the algebraic expression 7x represents (7 times x, 7 multiplied by x).
Strategies to help students overcome difficulties
Stepping in early and effectively is crucial to keep mistakes in solving equations from becoming insurmountable obstacles. Here are some tips.
- Make sure students clearly understand what it means to solve an equation. Solving an equation means finding the numerical value of the unknown so that, when it’s substituted back into the equation, the equality holds.
- Help your students understand the equal sign in an equation as a relationship of equivalence. Have them always check their solution after solving the equation, confirming that it’s the numerical value that keeps the equality true.
- Model the correct vocabulary. Every time you refer to the equation, make its meaning explicit. For example, instead of saying “7x,” say “7 times x,” or say “one x” instead of just “x.” It may feel a bit repetitive at first, but it’s a simple way to support your students’ understanding. You can also explore other related understandings: do students know that multiplying by 3/2 gives the same result as dividing by 2/3?
- Be more intentional when teaching students how to “undo” addition, subtraction, multiplication, and division. State explicitly when to add or subtract a term on both sides of the equation, or when to multiply or divide both sides by a number in order to isolate the unknown, etc.
As you can see, it’s all about making the implicit explicit! With this in mind, mathematical proofs are very useful for improving students’ understanding. To use them in class, we recommend this guide with visual, hands-on mathematical proof activities from ONMAT:
Mathematical proofs guide:
an explicit teaching strategy for Secondary
These are just some of the common difficulties that lead to mistakes in solving equations, but every student is different! That’s why, as teachers, it’s important to take time to observe how students work and at what points in the process they tend to struggle. Only then can we identify their difficulties and use strategies to help them overcome them.
