Gamification in Math Learning: Benefits & Challenges
Struggling to keep students engaged? Gamification, the use of game mechanics in learning, helps increase motivation, comprehension, and retention. The abstraction of concepts and the difficulty in linking them to real-life situations can generate disinterest, frustration and even anxiety in students. In this context, gamification has emerged as an innovative strategy that aims to transform the learning experience by integrating game elements into the educational process.
But what exactly is gamification, and what impact does it have on the teaching of mathematics? According to Deterding et al. (2011), it is defined as the use of game design elements in non-game contexts to enhance the participation, motivation, and engagement of individuals. In the educational setting, this involves not only the incorporation of game mechanics such as challenges, immediate feedback, points, and rewards, but also the creation of more dynamic and interactive experiences that encourage exploration and active learning.
Throughout this article, we will explore the existing evidence on the effectiveness of gamification in math education and see how interactive games like LearningLemur can benefit students’ learning.
Why Gamification Works: Key Benefits for Math Learning
Gamification has proven to be an effective tool for improving the learning experience in various disciplines, and mathematics is no exception. Its impact goes beyond making classes more entertaining; it influences key aspects such as motivation, understanding of concepts, and the development of cognitive and social skills. Below are some of the main benefits supported by research.
Increased Motivation and Engagement
One of the main challenges in teaching mathematics is to maintain students’ interest, especially when they perceive the exercises as repetitive or difficult. Gamification transforms these tasks into interactive and stimulating experiences. According to a study by Hamari, Koivisto, and Sarsa (2014), the implementation of gamified strategies has been shown to significantly increase motivation and academic performance in various disciplines, including mathematics.
Improved Comprehension and Retention of Concepts
Games allow students to interact with mathematical concepts in a hands-on, visual way, facilitating deeper understanding. Instead of memorizing formulas without context, students can experiment with problems in dynamic scenarios. A study by Wouters et al. (2013) indicated that gamified environments not only improve knowledge retention but also strengthen the ability to apply what is learned in different situations.
Cognitive and Social Skills Development
Beyond learning mathematical concepts, gamification promotes essential skills for critical thinking and problem-solving. Solving challenges within a gamified environment requires students to analyze situations, explore different strategies, and adapt their approach based on the results obtained. In addition, many gamified experiences include collaborative dynamics that foster communication and teamwork. It also encourages resilience by allowing students to learn from their mistakes without the fear of conven
tional academic failure.
Mathematics Anxiety Reduction
Math anxiety is a common problem that can affect students’ performance and attitude toward the subject. Presenting problems in a playful and less structured format can reduce the pressure associated with formal learning. According to Sailer et al. (2017), gamification helps to generate a more positive relationship with mathematics, allowing students to face challenges without fear of failure. Immediate feedback and reward systems reinforce individual progress, helping to build confidence and autonomy in learning.
Gamification Challenges: What Educators Need to Know
While gamification offers multiple benefits in mathematics learning, its effectiveness depends largely on how it is designed and implemented. Some key factors to consider for successful implementation are:
Designing appropriate content: Creating gamified activities aligned with curricular objectives requires time and planning on the part of educators.
Balancing Play and Learning: It is essential that the fun of the game does not overshadow the educational content, ensuring that students acquire the intended mathematical skills. To avoid this problem, it is key to design activities where progress in the game depends on mastery of the mathematical concepts, ensuring that the mechanics reinforce the content rather than distract from it.
Adaptability: Each student learns at their own pace and in a different way. An effective gamified approach must be flexible and allow for customization according to learners’ skill levels and learning styles.
Strategies for Effective Gamification in Mathematics Learning
To maximize the potential of gamification in math, educators and developers must carefully structure their approach. Here are some key strategies to ensure effective implementation:
Setting Clear Learning Objectives
Gamification must have a clear educational purpose to be effective. Before incorporating game elements, it is essential to define precise learning outcomes. Every challenge, reward, or interactive element should directly contribute to developing students’ understanding of mathematical concepts. When games are designed to align with curriculum goals, students remain engaged while making measurable academic progress.
Incorporating Interactive Games for Engagement
Engaging students through interactive math learning is one of the most effective ways to enhance learning. Digital platforms and apps provide adaptive learning experiences that offer students challenges suited to their individual skill levels. Features such as leaderboards, badges, and achievements create a sense of accomplishment, motivating students to persist in solving mathematical problems while reinforcing key concepts.
Providing Instant Feedback and Progress Tracking
Timely feedback is crucial in the learning process. One of the major advantages of gamification in math learning is the ability to provide instant feedback. When students receive immediate results on their performance, they can quickly identify mistakes, adjust their approach, and strengthen their understanding. Many digital platforms include progress-tracking tools, allowing both students and educators to monitor learning trends, address weaknesses, and celebrate achievements.
Encouraging Collaboration and Healthy Competition
Gamified learning is not just about individual progress—it can also foster teamwork and social interaction. Incorporating cooperative challenges and team-based problem-solving enhances mathematics learning by encouraging peer-to-peer discussions and shared strategies. At the same time, friendly competition through scoreboards and ranking systems can inspire students to push themselves further, increasing motivation and engagement in a healthy, constructive way.
Tools and Resources for Gamifying Mathematics Learning
With advancements in technology, numerous tools are available that support gamification in math. These resources offer diverse ways to create a more dynamic learning experience, helping students grasp mathematical concepts more effectively:
Digital Learning Platforms: Websites and apps that integrate math practice with games to teach mathematical concepts through fun and challenging activities.
Classroom Gamification Kits: Both physical and digital resources are designed to help teachers implement game-based learning strategies in their lessons.
Virtual and Augmented Reality: Immersive experiences that provide students with visual and hands-on interactions to better understand abstract mathematical ideas.
Customizable Gamification Plugins: LMS-integrated tools that allow educators to personalize their teaching by adding game-based elements to their curriculum.
Try Gamification in Mathematics Learning with Learning Lemur
The benefits of gamification in mathematics learning are clear: increased motivation, improved comprehension, and enhanced engagement. If you’re ready to experience the power of interactive games in education, explore Learning Lemur. Our platform allows educators and students to immerse themselves in engaging, game-based mathematics learning experiences.
Create a free account today and discover how gamification can revolutionize the way you learn mathematics!
Mathematical conjectures are riddles that challenge the human mind: seemingly simple problems that, despite having been verified in millions of cases, still lack a proof. They are seemingly intuitive questions that hide unexpected complexity.
The fascinating thing about conjectures is that, in many cases, they can be empirically verified i.e., verified by an enormous number of numerical examples. However, this verification does not satisfy mathematicians, who seek a theoretical proof, a solid proof that validates the statementin its entirety. Without such a proof, the conjectures remain open challenges.
The Collatz conjecture is a perfect example of this type of mathematical mystery. It wasproposed in 1937 by Lothar Collatz and its statement is as follows:
Choosing any positive integer we will apply the following steps:
➔ If the number is even, we will divide it by 2.
➔ If the number is odd, multiply it by 3 and then add 1.
➔ Repeat the process with the new number obtained.
The conjecture says that, no matter what number you start with, you will always get to number1. Once you get to 1, the process repeats indefinitely: 1 →4 →2 →1
Formally, this can be written as the function f=N →N defined as
Two illustrative examples of the Collatz Conjecture
Let’s look at a couple of simple examples.
Let’s take the number n=6 and apply the steps of the Collatz conjecture:
6 is even so we divide by 2: 6/2=3
3 is odd so we multiply by 3 and add 1: 3·3+1=10
10 is even so we divide by 2: 10/2=5
5 is odd so we multiply by 3 and add 1: 3·5+1=16
16 is even so we divide by 2: 16/2=8
8 is even so we divide by 2: 8/2=4
4 is even so we divide by 2: 4/2=2
2 is even so we divide by 2: 2/2=1
Once we reach 1, the process is repeated: 1 →4 →2 →1.
So we observe that starting with n=6 the conjecture is satisfied.
Let us now take the number n=21 and apply the steps of the Collatz conjecture:
21 is odd so we multiply by 3 and add 1: 21·3+1=64
64 is even so we divide by 2: 64/2=32
32 is even so we divide by 2: 32/2=16
16 is even so we divide by 2: 16/2=8
8 is even so we divide by 2: 8/2=4
4 is even so we divide by 2: 4/2=2
2 is even so we divide by 2: 2/2=1
Again, we arrive at number 1.
Both examples have reached the number 1, but in a different number of steps. In fact, although21 is a larger number, it reached 1 in fewer steps than 6. This conjecture has been tested for anincredible number of numbers, up to more than 2ˆ60 = 1.152.921.504.606.846.976 cases,without finding any counterexample. However, it still remains a mystery whether there is a number that does not satisfy the conjecture.
Graphical representations of the Collatz conjecture
The directed graph of orbits is a visual representation that facilitates the understanding of the behavior of numbers under the rules of the Collatz conjecture. In this graph, each number is represented as a node, and the connections between them show the steps followed by the conjecture process. When following the sequence of a number, the nodes are connected by arrows that indicate how the number is transformed at each step. Although even numbers are generally omitted from the representation for simplicity, the graph illustrates how the numbers “orbit” around certain cycles, such as 1 →4 →2 →1.
In the image above, we can see an example of the directed graph of orbits, where we have highlighted in black the numbers 3 and 21. The number 3 refers to the sequence of 6, since in this graph the even numbers are omitted to simplify the visualization.
We note that 21 quickly reaches 1, since it only passes through even numbers on its way, as wesaw in the previous example. On the other hand, the number 6 first becomes 3, then pass through the number 5, which will finally take it to 1 after several passes through even numbers.
It is also interesting to note that the number 9, although relatively small, follows a longersequence: it goes through the numbers 7, 11, 17, 17, 13 and 5, and finally reaches 1, after a total of 19 steps.
Below is a graph in which the X-axis shows the different initial integer values, while the Y-axis represents the number of iterations required for each number to reach the value 1.
In summary, the Collatz conjecture is an apparently simple problem that, to this day, still has no formal proof. Although it may seem a statement of little relevance or practical utility, this conjecture has applications in various fields, such as number theory, cryptography, algorithm analysis and artificial intelligence. In these fields, the study of complex sequences and their behavior under specific rules can offer valuable insights for solving larger problems and understanding mathematical patterns.
Have you been fascinated by the simplicity and mystery of the Collatz conjecture? Now it’s your turn: take the number 27 and start the sequence. How many steps will you need to get to 1?
If this challenge got you, don’t keep it to yourself! Share it with others who are curious aboutmathematical riddles and find out together who can solve the challenge the fastest.
Step-by-step solution of an Oxford access examproblem
Every year, thousands of students face the challenge of the admission exams to enter theprestigious University of Oxford, a process that tests not only their knowledge, but also theirability to solve under pressure. In particular, the math exam is known for its complexity, posingproblems of all kinds, from solving equations to questions of logic, advanced algebra, calculusand number theory. In this article, we will explore one of the problems presented in October2023.
The problem poses the following question: how many real solutions does the following equation have?
Do you dare to solve the equation before reading the complete solution in the blog?
At first glance, the statement seems simple: an equation with a single variable, where all the numbers involved are integers between 1 and 4. However, the real difficulty lies in correctlyundoing the squared parentheses, making sure to consider all possible cases.
A key detail in this type of equations is that, when squaring a number, the same result isobtained for both its positive and negative values. For example:
This occurs because squaring a number eliminates the negative sign. Therefore, when solving aquadratic equation we must take into account the two possible options: the positive and thenegative number that can generate that result.
One last observation: if we were to fully develop the parentheses, we would obtain an equation of the form xˆ16+…, which indicates that the equation is of degree 16 and, at most, could have 16 real solutions for x. Therefore, we cannot rule out any of the options presented to us aspossible answers.
Since for equations of degree 16 there is no systematic formula like Ruffini’s method for polynomials of degree 3, we will solve this equation by working the parentheses progressively,from the outermost to the innermost.
Specific example of how to solve an equation of degree 16
Let’s get on it!
Let’s start with the outermost parenthesis. If we define t as ((x²-1)²-2)²-3 , then we get t²=4. This gives us two possible values for t: t=2 or t=-2.
Substituting t by its original value in each case, we obtain the following results:
1. If t=2 then: ((x²-1)²-2)²-3 =2 ⇒ ((x²-1)²-2)²=5 and again, substituting r= (x²-1)²-2 we obtain that r²=5 . Let’s look at the two new cases that arise:
1.1 r=√5: substituting r for its original value we obtain (x²-1)²-2 =√5 ⇒ (x²-1)²=2+√5 and again, one last time, we substitute s=x²-1, we obtain that s²=2+√5 and observe the two results again:
When trying to calculate the square root to obtain the value of x, we would be taking the root of a negative number. As a result, the two values obtained will be imaginary numbers.
1.2 r=-√5 substituting r for its original value we obtain (x²-1)²-2=-√5 ⇒ (x²-1)²=2-√5 . Following the same reasoning as in the previous section, given that √4=2, it follows that √5>2 . This implies that 2-√5<0, and when trying to calculate its square root, we would obtain an imaginary number. Therefore, when we continue developing to obtain the value of x, this would also be an imaginary number.
2. If t=-2 then: ((x²-1)²-2)²-3 =-2 ⇒ ((x²-1)²-2)²=1 and again, substituting r= (x²-1)²-2 we obtain that r²=1 . Let’s look at the two cases:
2.1 r=1: substituting r for its original value we obtain (x²-1)²-2 =1 ⇒ (x²-1)²=3 and again, one last time, we substitute s=x²-1, we obtain that s²=3 and observe the two results again:
2.1.1 s=√3 and substituting s for its value, we finally obtain: x²-1=√3 ⇒ x²=1+√3 and therefore
2.1.2 s=-√3 substituting s for its original value we obtain x²-1=-√3 ⇒ x²=1-√3 . This subtraction is negative, since √2≈1,41 and therefore, √3>1,41 with which we deduce that 1-√3<0 , and when trying to calculate its square root, we would obtain an imaginary number.
2.2 r=-1 substituting r for its original value we obtain (x²-1)²-2 = -1 ⇒ (x²-1)² = 1 and again, one last time, we substitute s=x²-1 and we obtain that s²=1 and observe the two results again:
2.2.1 s=1 and substituting s for its value, we finally obtain: x²-1 = 1 ⇒ x²=2 and therefore
2.2.1.1 x=√2 which is a real solution.
2.2.1.2 x=-√2 which is a real solution.
2.2.2 s=-1 and substituting s for its value, we finally obtain: x²-1 = -1 ⇒ x²=0 and therefore the only possible solution is x=0 .
Let’s review all the possible real values we have obtained:
This gives us a total of 7 real solutions for x, so the correct answer is option (c).
Did you get your answer right?
Drawing this equation with our graphing tool is a simple and effective way to check the number of real solutions. Looking at the graph, you can clearly see how the curve interacts with the x-axis, confirming the previous result. Below, we show you the graph to see for yourself:
Recently, we had the exciting opportunity to showcase our approach at BETT 2025: Enhancing Open-Ended Math Questions with Instant Feedback to teachers, publishers and educators alike at the conference in London. Feedback plays a crucial role in guiding learning, and at the event, we focused on how instant feedback in open-ended math questions can dramatically enhance students’ mathematical understanding, all the while assisting teachers and making their day to day more efficient.
Unlike traditional multiple-choice problems, open-ended mathematics questions require deeper thinking as they are questions that a student can freely answer without a predefined format, making personalized feedback more vital. In this article, we will explore the significance of instant feedback in open-ended math questions and discuss how different types of feedback—corrective feedback, confirmatory feedback, and suggestive feedback—can help students improve.
Why Instant Feedback Matters in Open-Ended Math Questions
The Power of Feedback in the Learning Process
Feedback is integral to the learning process, helping students refine their thinking and improve their understanding. This is especially true for open-ended mathematics questions, where there is often no one right answer. In such cases, instant feedback in open-ended math questions can make a significant difference in guiding students toward the correct solution.
During BETT 2025, we discussed how instant feedback in math questions not only helps students correct mistakes quickly and builds their confidence but how it also assists teachers in the class environment making the teaching and learning process much more efficient.
Types of Feedback for Open-Ended Math Questions
Different types of feedback are required depending on the student’s needs and the nature of the question. The three main types of feedback we focused on at BETT 2025 are suggestive feedback, confirmatory feedback, and corrective feedback.
Suggestive Feedback in Math Questions
Suggestive feedback in math questions guides students to discover the solution themselves. Instead of providing the correct answer outright, suggestive feedback encourages independent problem-solving. For example, if a student suggests “2 and 8” as the two numbers whose sum is 10 and product is maximum, suggestive feedback might be: “What happens if you try numbers closer together?”
Confirmatory Feedback in Math Questions
Confirmatory feedback in math questions is designed to reinforce correct answers. When students solve problems correctly, feedback like “That’s correct, well done!” helps to affirm their understanding and boosts their confidence.
In the case of open-ended math questions, confirmatory feedback is invaluable for ensuring students stay motivated, especially when they are tackling complex problems.
This type of feedback fosters critical thinking and encourages exploration, making it an essential tool for developing problem-solving skills in open-ended math questions.
Corrective Feedback in Math Questions
Corrective feedback in math questions identifies mistakes and provides the correct answer. For example, if a student incorrectly solves x^2=4 as x=4, the corrective feedback would be “The correct solution is x= +-2”.
While effective for addressing misunderstandings, corrective feedback should be used in moderation, as over-reliance on it can hinder the development of problem-solving skills.
The Teaching Cycle: A Framework for Feedback Application
At BETT 2025, Mrs. Brook, a fictional high school math teacher, shared her teaching cycle, which integrates corrective, suggestive, and confirmatory feedback seamlessly. Her cycle is designed to maximize learning outcomes by aligning feedback strategies with the different stages of teaching.
The Four Stages of Mrs. Brook’s Teaching Cycle
Delivering the Content
Mrs. Brook begins her teaching by introducing and explaining the content to her students. This step ensures all students have a baseline understanding of the topic.
End-of-Class Knowledge Validation
Following the lesson, Mrs. Brook conducts an end-of-class knowledge check. This is where corrective feedback plays a crucial role. For example, if a student misunderstands a concept or calculation, Mrs. Brook provides immediate corrective feedback to clarify misconceptions.
Recommending Practice at Home
Mrs. Brook encourages her students to practice independently. At this stage, she often employs suggestive feedback to guide students without directly giving them answers, helping them to think critically and explore solutions. Additionally, confirmatory feedback is used to reinforce correct solutions and build student confidence. When students solve problems accurately, Mrs. Brook provides positive reinforcement such as “Great job!” to motivate them and affirm their understanding.
Assessing Students’ Knowledge Levels
After students have had the opportunity to practice, Mrs. Brook assesses their progress to identify areas of improvement. During this phase, she focuses on encouraging students to reflect on their learning journey and strive for continued growth. She fosters a positive environment by recognizing their efforts and offering motivation to help them stay engaged and confident in their abilities.
This cyclical model of teaching and feedback showcases how structured feedback enhances learning and ensures students’ needs are addressed at every stageTech.
Wrapping up: Embracing the Future of Math Education with Wiris
At WIRIS, we’re committed to supporting educators with the tools they need to provide effective feedback. Our Learning Lemur and WirisQuizzes products ensure that instant feedback in open-ended math questions is accessible and customizable, helping students succeed in their math learning journey.
As technology continues to evolve, the integration of instant feedback into open-ended math questions will only become more critical. We look forward to the continued evolution of math education, driven by personalized learning and real-time feedback.
