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Why do students start struggling with math in middle and high school?
How does struggling begin?
Many kids do just fine with math in elementary school, so it can be a surprise to parents when things take a turn in middle school. At first, a student might fail one math test. Then another. They might start blaming circumstances—lack of time, noisy classmates, or distractions that made it hard to focus. Next, the blame might shift to the teacher. Eventually, they might say, “I’m just not a numbers person.” In the worst cases, students might even start boasting about being bad at math, convincing themselves and others that math is pointless. So, what went wrong? And where exactly did things start to fall apart?
Why does the struggle start?
Elementary school math feels real and straightforward—counting apples, sheep, and friends. But middle school math dives into the abstract: denominators, equations, and functions. This shift from concrete to abstract math is challenging for many kids, and navigating it successfully requires extra support. Teachers understand this and are equipped to help. But with a packed curriculum, large classes, and a fast pace, they often can’t provide the personalized help every student needs. Now, imagine a student misses a key lesson—maybe they were sick or just had an off day. If those gaps aren’t filled, they’ll try to patch things together themselves. But how do they go about it?
Strategies students use to overcome struggles
Using intuition
In everyday situations, intuition can be helpful. But in math and science, things are often counterintuitive. Take percentages, for example: if the price of an item increases by 50% and then decreases by 50%, many students intuitively think the price returns to its original value. However, this isn’t correct. When intuition fails, students often resort to other strategies like rote memorization.
Rote memorization
Rote memorization is quite a typical approach. For example, when dividing fractions, students often memorize the rule: divide by a fraction by multiplying by its reciprocal. For instance: \[ \frac{1}{3} \div \frac{1}{9} = \frac{1}{3} \times \frac{9}{1} = \frac{9}{3} = 3 \] It’s much easier to memorize this rule than to understand why it works. And when memorization seems like the easier path, many students take it without hesitation. However, in the long run, neither intuition nor rote memorization is a sustainable solution. Up next, we’ll explain why.
Why these strategies don’t work
Relying on memorization creates a habit:
“If I don’t understand something, I’ll just memorize it.”
This approach might work through 7th or 8th grade, or even a bit longer, but eventually, it becomes unsustainable:
No one can memorize all of math.
This is also why students often forget what they’ve “learned” so quickly. In reality, they haven’t truly learned it—they’ve only memorized it. Often, the material is forgotten right after the test. And when that happens, here’s what comes next:
What happens next?
When a student forgets everything they just memorized, they have to learn every new math topic as if starting from scratch, without building on previous knowledge.
This forces them to put in enormous effort—yet with very little progress. This creates a vicious cycle. The more they struggle, the more they rely on memorization. With a fast-paced curriculum that moves quickly from one topic to the next, some students soon find themselves unable to follow along at all. So, how can they break out of this cycle?
How to stop struggling and perform better?
Take a step back
As we mentioned earlier, it’s essential to fully understand each topic before moving forward—not just memorize it. But what if gaps in understanding have already formed? The solution is straightforward: take a step (or several) back and figure out where things started to go wrong. That’s exactly why we created this article.
Identify problematic math topics
This article highlights common math topics that students often struggle to grasp fully. When these topics are not understood, they lead to confusion and reliance on ineffective strategies, like rote memorization. Share these topics with your middle or high schooler and ask them to answer related questions. If they answer correctly, great! It’s likely they understand the topic well. If not, the topic might need revisiting and reinforcing.
Address the problems
Once a weak spot has been identified, have your child review the correct answers along with an explanation of their confusion. We’ve included why these misunderstandings can cause long-term issues and how they might impact future learning, professional skills, and even daily life. And most importantly, don’t forget to encourage and motivate your son or daughter to tackle these challenges. Up next, we’ll share some tips on how to keep them motivated.
How to motivate your middle or high schooler to work on it?
Why do we learn math?
As math becomes more abstract, students often ask, “Why do I need to learn this?” Understanding the “Why” behind math is key to keeping students motivated. At DARTEF, we take this question very seriously and are dedicated to providing clear, thoughtful answers. We believe the best answer to “Why?” is clear and simple:
Because math is essential in both professional and daily life.
At DARTEF, we know exactly how to explain this.
How is math used in professional and daily life?
Math plays a role in nearly every profession. Even jobs that don’t seem math-related—like nursing, law, and psychology—often require specific math skills. At DARTEF, we create articles and videos that show how professionals from all walks of life use math in their work. So, the next time your middle or high schooler says, “I don’t need math because I’m going to be a nurse, lawyer, or psychologist,” share our articles or videos with them. Articles are free to access on our blog, and our videos are available for a small fee. Why subscribe?
List of problematic math topics that make students struggle
Table of Contents
Introduction
Math topics are organized by grade for easier navigation. However, keep in mind that a topic learned in, for example, grade 6, can often affect the understanding of topics in later grades, such as grades 8 or 9. Therefore, we recommend making sure that foundational topics from earlier grades are also well understood. For instance, if your child is in grade 9, ensure they grasp the key concepts from earlier grades before moving forward. Please note that this is a living article, and we are continually updating it (the latest update was on December 10, 2024).
Elementary school
Division
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Share this quiz with your kid:
- Partition (sharing): 8 is divided into two equal parts, each part being 4.
- Measurement (repeated subtraction): 8 ÷ 2 represents how many times 2 can be subtracted from 8 (i.e. 8-2-2-2-2=0, 4 times).
Common confusions:
Students often only recognize the first way (partitioning) and are surprised to learn about the second way.
Why these confusions happen:
This confusion arises because partitioning is intuitive and connects to everyday experiences, such as dividing 8 candies among 4 friends.
Why it matters:
When students rely too heavily on intuition, it can lead to issues in future learning, professional life, and even in daily activities.
In further studies: Understanding repeated subtraction is key to grasping more advanced concepts, such as dividing fractions. For example, 1/3 ÷ 1/9 = 3, because you can subtract 1/9 from 1/3 three times.
In professional life: Repeated subtraction is frequently used in logistics. For example, how many trucks are needed to move 10,000 tons of asphalt, if each truck can carry 25 tons? 10,000 ÷ 25 = 400.
In daily life: Repeated subtraction is helpful in practical tasks. For instance, if you’re building a 120-foot fence and want to space the fence posts 8 feet apart, you’ll find there will be 15 spaces (and 16 posts).
How to motivate a student?
Want to spark your teen’s interest in practical uses of division? Check out this section on making math relatable and engaging!
Geometry
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Share this quiz with your kid:
Square.
Common confusions:
Sometimes, students may mistakenly call the resulting shape a rhombus or diamond. In reality, when a square is rotated by 45°, it remains a square. Rotation does not change the formal properties of the square.
Why these confusions happen:
This confusion arises because students often rely on visual memorization of how shapes appear, while paying less attention to their formal properties.
Why it matters:
Relying too much on visual memorization can lead to challenges in future studies, professional life, and even in daily situations.
In further studies: Students who depend on visual memorization might struggle when they encounter non-standard representations of shapes on exams.
In professional life: In fields like medicine, relying on typical imaging appearances can lead to diagnostic errors. For instance, a radiologist may misdiagnose a disease when its presentation doesn’t “look” like the textbook example, missing subtle but crucial signs.
In daily life: In navigation, someone who memorizes a map’s appearance without understanding cardinal directions or scales might struggle to navigate in unfamiliar areas. Real-world navigation often involves adapting to different perspectives and reorienting oneself based on landmarks—something memorization alone cannot address.
How to motivate a student?
Looking for ways to keep your teen curious about shapes and their properties? Check out this section on fostering curiosity in math. ↑
Source: [1]
Grade 6
Equalities
![On the green chalkboard, it is written: 'Fill in the missing number: 8 + 4 equals [blank] plus 5.'](https://dartef.com/wp-content/uploads/2024/11/equality1.jpg)
[Click the picture to enlarge]
Share this quiz with your kid:
7
(because the left side of the equation must always equal the right side.)
Common confusions:
Sixth graders sometimes misunderstand the equality sign (=) and see it as “and the answer is.” This confusion can lead them to give answers like 12 (thinking 8 + 4 = 12) or 17 (thinking 8 + 4 + 5 = 17).
Why these confusions happen:
In elementary school, students often memorize the visual pattern of “digit + digit = answer.” When they encounter equations in higher grades, this visual memory can conflict with their understanding of equality and algebraic equations, where both sides must balance, not just the addition of the numbers.
Why it matters:
- In further studies:
Understanding equalities is crucial for grasping concepts like systems of linear equations. When substituting expressions, it’s essential that equality is maintained to ensure the solution is accurate. - In professional life:
A key example of equalities in the financial sector is the basic accounting equation: Assets = Liabilities + Equity. This equation helps companies ensure their financial records are balanced and accurate, forming the foundation of financial reporting and analysis. - In daily life:
In health and nutrition, understanding equalities can help in planning a balanced diet. For instance, it allows for precise calculations of calories and helps adjust meal portions according to individual nutritional needs, contributing to healthier living.
How to motivate a student?
Looking for ways to keep your teen curious about the equality sign and balancing equations? Check out this section on fostering curiosity in math. ↑
Source: [3]
Percentages
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Share this quiz with your kid:
No. The correct interpretation is: “20% of ALL RUNNERS are males from Town C.”
Common Confusions:
Students often mistakenly agree with the incorrect statement “20% of Town C males are runners.”
Why does this confusion happen and how to fix it?
This confusion is common among students learning about percentages. The issue often stems from misinterpreting the graph. The incorrect statement “fits” with the visual cue of 20% on the graph, leading students to choose it without fully analyzing the data.
- Start by saying: “Don’t rush to choose the answer that seems to fit well.”
- Ask: “What does the graph actually show?”
The correct answer here is: “The distribution of runners from towns A, B, and C.” - Clarify the “whole” being represented.
Stress that percentages depend on what constitutes the “whole.” In this case, the whole is “all runners”, not “Town C males”.
Why it matters:
- In Further Studies:
Percentages and ratios are essential in many subjects, including physics (e.g., efficiency calculations), chemistry (e.g., concentration percentages), and biology (e.g., population studies). - In Professional Life:
Accurate data interpretation is vital in almost every job. For example, in marketing, avoiding errors like the Simpson’s Paradox is important when analyzing customer behavior trends. - In Daily Life:
Understanding percentages and graphs is essential for daily tasks like voting on policies, where graphs might show the distribution of budget allocations or public opinion percentages.
How to motivate a student?
Looking for ways to keep your teen curious about percentages and graphs? Check out this section on fostering curiosity in math. ↑
Sources: [4] and [5].
Subtracting negatives
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Share this quiz with your kid:
When you subtract −1, it’s like removing a negative value. Taking away −1 is the same as adding 1, because you’ve removed the “negative” part.
Common Confusions:
It’s just a rule!
Some students, unsure of why this happens, simply say, “It’s just a rule,” without understanding the logic behind it.
Why does this confusion happen and how to fix it?
While subtracting a negative makes logical sense, some students choose to memorize the rule “Minus and minus gives a plus” because it’s simpler to remember than to fully grasp the concept. To help students understand, practical examples like “removing a debt” or a “temperature rebound” are effective in illustrating the core principle of subtracting negatives.
Why it matters:
- In Further Studies:
When students later learn about multiplying or dividing negatives, they might continue to rely on memorization. This habit can make it harder for them to grasp why “Minus times minus gives a plus” actually works. - In Professional Life:
Working with negative numbers is essential in many professions. They appear in accounting (balancing budgets), construction engineering (calculating loads), telecommunications (managing signal strength), and countless other fields. Understanding how to operate with negatives accurately ensures reliable results and prevents costly mistakes. - In Daily Life:
Grasping how negative numbers work is key for everyday tasks, from managing finances to interpreting weather forecasts. For instance, if the forecast mentions that a “drop of -5°F will rebound,” it means the temperature will rise by 5°F. Without a solid understanding of negatives, simple calculations and interpretations can become unnecessarily confusing.
How to motivate a student?
Looking for ways to keep your teen curious about subtracting negatives? Check out this section on fostering curiosity in math. ↑
Zero
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Share this quiz with your kid:
8, 110, and 0.
Common confusions:
Students often identify only 8 and 110 as even numbers, excluding 0. In reality, 0 is even because it is divisible by 2 (0 ÷ 2 = 0).
Why these confusions happen:
This widespread misunderstanding stems from how zero is introduced in elementary school, often described as “nothing.” This leads to the false belief that “nothing” cannot be divisible, and thus, 0 is neither even nor odd.
Why it matters:
Understanding zero as a number, not as “nothing,” is crucial for further studies, technology, professional applications, and daily life.
- In further studies: In trigonometry, recognizing that a triangle’s leg can have a zero length is important, e.g., cos 90°=0.
- In professional life: In fields like information technology, zero often represents “FALSE” in programming (opposite to “TRUE”), while in accounting, zero indicates a balanced budget, not an absence of value.
- In daily life: Zero degrees on a temperature scale (e.g., 0°F or 0°C ) represents a specific point, not “nothing”.
How to motivate a student?
Looking for ways to keep your teen curious about zero and its properties? Check out this section on fostering curiosity in math. ↑
Source: [2]
References
- Rafiah, H., & Ekawati, A. (2017, May). Misconceptions of the students with high mathematical creative thinking level in solving the geometric shapes problems. In 5th SEA-DR (South East Asia Development Research) International Conference 2017 (SEADRIC 2017) (pp. 155-158). Atlantis Press.
- Arai, N. H., Sugawara, S., & Matsuzaki, T. Misconception about Zero: How Difficult it is to Overcome. The 15th International Congress on Mathematical Education Sydney, 7-14 July, 2024
- Falkner, K., Levi, L., & Carpenter, T. (1999). Early childhood corner: children’s understanding of equality: A foundation for algebra. Teaching children mathematics, 6(4), 232-236.
- Schield, M. (2006). Statistical literacy survey analysis: Reading graphs and tables of rates and percentages. In Proceedings of the Sixth International Conference on Teaching Statistics.
- Rakes, C. R. (2010). Misconceptions in rational numbers, probability, algebra, and geometry. University of Louisville.