10 key points to improve mathematics teaching
Introduction
During primary school, mathematics remains the favourite subject of most children. However, upon reaching secondary school, it becomes the bane of many, even a trauma for some. How did we reach such a situation? What is our educational system missing that creates such a lack of interest, not to mention the crisis of teaching this beautiful discipline?
In February 2018, a report on mathematics teaching in France was presented to the French Minister of National Education, Jean-Michel Blanquer. The report aimed to assess practices, identify strengths and weaknesses, and propose concrete solutions to restore the place and appeal that mathematics currently lacks. The project was led by Cédric Villani, Fields Medalist (equivalent to the Nobel Prize in our field) and Charles Torossian, General Inspector of Mathematics.
The discussion was led by various professionals, including math teachers and manual creators, who focused on specific topics such as the challenges faced in teaching math, the fundamentals of calculation and automation, and the effectiveness of different teaching methods. The role of resources and books in enhancing teaching practices was also discussed.
As a math enthusiast, I found these discussions fascinating and am eager to share my passion for the subject. One particular point that caught my attention was the positive impact of making mistakes in the learning mechanism, which eventually leads to success. This was highlighted in a report titled “21 Measures for Math Education.”
Here are the top 10 most important points to revitalizing mathematics education.
1 — Considering the positive role of mistakes
To encourage students to engage in mathematical problem-solving, removing the stigma around making mistakes is important, as it is a fundamental aspect of the research process.
The strategy involves embracing mistakes during the learning process and using them as a tool for student growth and development.
One option we have implemented involves distributing an error report to students after each exam, which lists the mistakes and areas where they can improve. This report also includes positive feedback for questions that were correctly treated.
Another option is to generate individualized error reports for each student after each exam, which allows them to track their progress and adjust their teaching accordingly.
Homework given to students for evaluation purposes should be optional and not graded, as some students tend to copy from exemplary papers.
This approach has been successful in higher education, where students understand that it is an opportunity for personal production and errors are identified rather than penalized.
Differentiated homework assignments focusing on deepening concepts or foundational knowledge should also be considered.”
2 — Teaching historical errors
Teachers should consider incorporating history into math lessons, as they do in other courses such as economics and philosophy. Students are fascinated to learn that mathematical thinking has a rich history of success, mistakes and failures from even the greatest spirits and that it is still a thriving field today.
Unfortunately, History is only briefly mentioned in school when introducing Thales and Pythagoras’ theorems in middle school. However, there are plenty of opportunities in the curriculum to engage students. In high school, the introduction of complex numbers is a prime example. Students are amazed to learn that solving algebraic equations was a concern for the Babylonians and that the d’Alembert-Gauss theorem, which sheds light on these issues, only dates back to the 18th century.
It’s amusing to narrate to them that Gauss discovered an error in the proof, which wasn’t completely resolved until Galois in the 19th century. Even in elementary school, children can be taught some epistemology by providing historical context linked to their learning without overwhelming them with knowledge. Finally, while it’s not about ranking topics in mathematics, the 2000s curriculum favoured probability and statistics over geometry.
3 — Granting geometry the importance it deserves
Geometry is often treated as a separate subject towards the end of the year, and even in science-focused classes, it is often given less importance. However, it is crucial for students to develop their reasoning skills and learn how to construct mathematical arguments.
Unfortunately, many students lack confidence in geometry due to limited exposure in middle school. Based on my experience teaching middle school, I have seen the importance of using visual aids and taking the time to develop students’ deductive reasoning skills. For this purpose, Geometry provides a tangible way to test mathematical concepts and develop arguments.
Encouraging students to reclaim the use of forgotten tools such as the square and compass, or even to consider using an unmarked ruler and compass only, helps them think critically about geometric concepts. In my experience, students often struggle to connect the trigonometric circle with equilateral triangles and angles of 60 degrees, even though they learned this in elementary school.
Ultimately, teaching geometry can introduce students to the basics of axiomatic. The mathematical theory remains valid as long as the deductive structure holds true.
4 — Taking profit from the Singapore method
As parents, we became aware of the approach to mathematics used in the Singapore method when we had to homeschool our eldest son in CE1 for six months.
We searched for reliable and clear resources to teach him mainly French and mathematics at home and discovered the manuals based on what was practised in Singapore, published by La Librairie des écoles.
The didactic approach impressed us with its clarity, ease of implementation despite the fairly strong demands, the coherence of its progression, and above all, our children’s autonomy in approaching the exercises.
As emphasized in the report, it is important for parents to be able to support their children and understand the expectations; the more accessible the resources are to parents, the more they become involved in their children’s learning.
Looking at student work over time, it becomes clear that this has not really been experienced in our school system. The constant changes in programs, which are sometimes seen as arbitrary, and the lack of ongoing evaluation have prevented us from taking a step back on certain practices to optimize them.
A practice is only useful if it benefits the majority, but how can we measure its impact if we only receive feedback at the classroom level?
Furthermore, the fact that students are guided very explicitly, especially through short statements without distracting illustrations, reassures them and allows them to focus on what is essential.
I have seen too many elementary school books resembling comic strips where the instructions are difficult to find, which can be stressful for children who are lost in the task.
By feeling confident in approaching mathematical concepts, students from a young age can trust themselves and follow their intuition. With the Singapore approach, introducing the four operations at the very beginning of elementary school allows students to become familiar with all calculation mechanisms early on.
5 — Prioritizing calculative exercises
Mastering basic mechanisms, including calculation, to solve mathematical problems is essential. Additionally, numbers provide an excellent opportunity for practical manipulation and understanding of mathematics, serving as a gateway to more abstract concepts.
Unfortunately, many students cannot progress to complex reasoning as simple numerical manipulations hinder them. Even the most proficient students in the subject are often stymied by basic algebraic considerations.
They often struggle with fractions, square roots, and powers in mathematics. Just as one must know the basics of grammar and conjugation to write correctly in French, it is essential to have a solid foundation in mathematical calculations to enjoy solving mathematical problems.
It’s important to spend time in class exercising calculation regularly. This exercise helps them develop their skills and often becomes a fun game. Consistency is crucial when practising these skills to create automaticity.
Providing a database of exercises designed to develop calculation skills for teachers at every level of education could be a helpful tool. Enjoying mathematics is a crucial factor in success and should be emphasized.
6 — Offering enjoyable mathematics experiences.
Finding pleasure in math often involves facing a challenge. When one experiences the joy of pushing oneself beyond obstacles, one craves the opportunity to relive that feeling. This can be achieved through mathematics.
While we readily accept this concept in sports, it is not often associated with pleasure in education, especially in our discipline. It is important for students to take pride in their accomplishments, even if they make mistakes or experience occasional failures, as long as they ultimately achieve their goals.
Additionally, practising mathematics outside the classroom is crucial, particularly through clubs, games and competitions. It is essential to keep the discipline alive beyond the classroom, mix up the participants, and bring together students of different ages to work on common projects.
I had the opportunity to lead a mathematics workshop for four years (called “Method of Learning Mathematical Theories by Pairing Schools for a New Approach to Knowledge”) and was impressed by the enthusiasm and motivation of the students.
Students learn to explore a topic over an extended period of time without the pressure of immediate results. They are encouraged to make mistakes and try different approaches, even if they don’t lead to success.
Experienced students can help younger ones understand complex concepts by breaking them down into simpler terms. Probability and geometry can be taught through hands-on activities and experimentation. Students present their findings orally, which is challenging but ultimately rewarding. They enjoy the process of discovery and take pride in their collective achievements.
The fact that students willingly devote their free time to exploring mathematics through ateliers is proof that they are eager to learn. It’s worth noting that those who excel in math aren’t necessarily the most involved in these activities, so it’s a great way to value different skill sets.
Saving space and time for these initiatives within schools is crucial, and it’s important to encourage as many students as possible to participate. Activities like these demonstrate that learning can be fun and enjoyable, not just a chore.
7 — Setting explicit expectations
Ensuring the coherence of effective methods by clearly indicating the intended learning outcomes at a given level is absolutely essential for both students and teachers.
For that to happen, we distribute a comprehensive but concise checklist of expectations regarding knowledge and skills based on the Official Bulletin program at the beginning of each chapter.
This approach also applies to high school, even if students are expected to have a certain level of autonomy and take charge of their learning. This also makes them feel more confident when they have covered the necessary concepts.
Regrettably, this process often remains on the teacher’s side, who sets precise requirements but doesn’t communicate them to students. However, clearly stating expectations reassures students and involves parents in their children’s education.
8 — Furnishing learners with a thoroughly crafted piece of writing.
Selecting appropriate written materials can make all the difference in a student’s understanding of the subject. Math is often challenging for many students, and the right resources can provide them with the necessary tools to succeed.
For efficiency purposes, we select concise, comprehensive, challenging, yet approachable materials. We also opt for distributing a complete copy of each chapter at the beginning of the course, which offers many advantages :
- Firstly, students are not constantly absorbed in the task of copying from the board, which can become mechanical, and they may not understand the content.
- Secondly, students can take their own notes based on preliminary information that is not on the document, illustrations, and additional information given by the teacher orally or on the blackboard.
- Thirdly, students of different levels can benefit, as the more alert ones have the entire course and can anticipate the following concepts, while those who are struggling always have something to refer back to.
The format of the copies is quite unique as it consists of tables separated into lessons, methods, comments and exercises.
The comments column allows for detailed treatment of an example, stating a theorem in more common terms, or even getting into a demonstration.
The exercises are designed to target specific skills but progress gradually, and the final ones in the chapter incorporate all of the themes covered in the various sections.
9 — Establishing meaningful connections between disciplines
More often than not, physics and mathematics curricula are not well-coordinated, particularly in terms of timing. It is challenging to grasp concepts such as force or displacement without a strong understanding of vector objects and scalar products. In chemistry, decimal logarithms are introduced very early in the year, and determining trajectories in kinetics requires a solid grasp of the concept of primitives with initial conditions.
A concerted dialogue is required to harmonize the progression. When real discussion beforehand is achieved, disciplines can feed off of each other, avoiding unnecessary difficulties for students who may not always express their concerns to teachers when they have not yet covered the necessary tools for a good understanding of the physics course.
Regarding the teaching of mathematics in France, some exercises lack significance in terms of content or learning. Through exchanging ideas on linear regression, a colleague and I realized that our course could be improved by combining our practices to benefit our students.
Mathematics should not solely be viewed as a set of tools by other disciplines. Instead, it can be used to present intriguing scenarios that connect various fields of knowledge to students.
10 — Share media and practices
This point is not addressed in the reports but seems fundamental to me.
Though we’ve been instructing modern mathematics for decades, and considering that the syllabus doesn’t change fundamentally with each reform, we’ve not yet implemented any organized strategy for sharing teaching resources in a streamlined manner.
At a high level, it would be beneficial to have a shared space for collaborative contributions from all individuals. Having a collection of resources readily available would alleviate the burden for novice educators who would otherwise have to begin designing lessons from scratch. This initiative should be generalized in both space and time.
At a lower level, fellow mathematics teachers talking about the difficulties encountered by their pupils is common, however, it is quite rare for us to go into detail about practices and lessons. Most likely a matter of humility. But, from my experience, sharing our tips and tricks is frequently fruitful and can lead to concrete ideas we wouldn’t have thought of spontaneously. This was the case when I started writing up Error report after a class assignment.
Conclusion
In conclusion, revitalizing mathematics education requires a multifaceted approach that embraces the historical, practical, and innovative aspects of teaching.
By acknowledging the value of mistakes as learning opportunities, we can transform the classroom into a place of exploration and growth. Integrating historical errors into the curriculum not only adds depth to the learning experience but also humanizes the subject, revealing the dynamic journey of mathematical discovery.
Geometry, an often-neglected cornerstone, must reclaim its rightful place in the curriculum, providing students with the means to strengthen their deductive reasoning and spatial visualization skills. Moreover, adopting successful international teaching methods, such as the Singapore approach, can offer clarity and a structured progression that empowers both students and parents alike.
Calculative exercises should be prioritized to ensure that students develop a solid computational foundation, which is necessary for more advanced mathematical reasoning. However, it’s not all about rigor and discipline; mathematics must also be enjoyable. Creating pleasurable mathematical experiences through clubs, games, and collaborative projects can spark a lifelong passion for the subject.
Furthermore, setting explicit expectations and providing students with well-crafted written materials can significantly enhance their understanding and mastery of mathematical concepts. As educators, parents, and policymakers, we must commit to these principles to foster an environment where mathematics is not merely a subject to be learned but a fascinating world to be discovered.
By implementing these 10 key points, we can hope to inspire a new generation of mathematicians who view the subject not as a trauma but as a treasure trove of challenges and triumphs.
