It is difficult for many students and pupils to perceive mathematical calculations in the context of dry numbers and memorized formulas. As a result, general mathematics education among the population suffers and people have difficulty obtaining degrees in areas directly related to the subject. However, in the modern world, teachers are moving away from the old teaching practice, applying innovative technologies and visualizing the process. Thus, students are easier to understand mathematics and they are more confident in learning the subject.

Prior to reading Jo Boaler’s book, Mathematical Mindsets: unleashing students’ potential through creative math, inspiring messages and innovative teaching, I already knew some of the basics of the subject. First of all, modern approaches to teaching a subject consist in the visualization of content for easier assimilation. The brain works better when a person visually sees possible ways to solve a problem. Because of this, the constant outlining of examples in the context of mathematics increases the chances of a more successful understanding.

Moreover, creative approaches in mathematics play an important role and bring variety to the process. Interactive activities relax students and allow them to rest by changing activities. Excursions, game simulations of tasks and open discussions are aimed not only at consolidating the material, but for the most part, at learning new information. At the same time, the student does not feel outside the context of the lesson, but inside it, which encourages them to independently search for solutions. As a result, they get the experience of active learning, which is no less important than their passive learning.

In addition, innovative methods in the study of mathematics are aimed at applying a communicative approach. The essence of obtaining knowledge in its application lies in the fact that learning is of an activity nature. Constant dialogues and open debates help students become interested in the topic and defend their position in solving problems. First of all, this is due to the widespread introduction of collective approaches to work. The teacher acts as a cooperator who, along with the students, is looking for a solution to a mathematical problem and puts forward his own assumptions. The ultimate goal is determined by the fact that students can calmly discuss points that they do not understand and express their opinion on mathematics lessons.

Moreover, before reading the book, I knew about the integrated approach, in which there is an interpenetration of mathematics and other sciences. Such lessons bring novelty and practicality to the ordinary school system, which leads to the identification of a number of advantages. They increase people’s motivation due to the originality of the presentation of the material and form a cognitive interest, which is subsequently directed to the desire for self-education. In this case, the teacher and student are at the level of subjective relations and can equally approach creative solutions to the problem.

However, after reading the book, knowledge expanded and, in this regard, the main conclusions can be drawn. First of all, it is the understanding that the so-called “mathematical brain” does not exist (Boaler, 2022). Students can tailor their efforts in both liberal arts and technical subjects. In this case, the correct setting of the concept for growth and willpower directed to hard work for a certain time is important. At the same time, differences in the brain at birth are still observed and people can show certain inclinations towards different disciplines. Nevertheless, more essential is the hard work that students put in during their lives.

Another takeaway was the consideration of the power of mistakes and their impact on the learning process. Wrong decisions stimulate the brain to find the right solution and students develop in the subject. In this case, a lot depends on the teacher, who can tell people that mistakes are a very positive learning experience (Boaler, 2022). As a result, the focus shifts to the ability to try new ideas and the desire to find the answer to a mathematical problem.

Additionally, students do not need to be assessed on their ability to solve problems. The propensity for mathematics and its application in real life have a huge difference, and this should be considered in the learning process. Thus, despite poor performance in the classroom, they can successfully apply knowledge outside the walls of the educational institution. Moreover, it can stimulate them to be more successful in the classroom and the desire to acquire new knowledge.

One of the main conclusions was the encouragement of intuition and freedom of thought in students. Many people who have studied mathematics in the past years and decades have taken the standard curriculum. They solved problems according to given formulas, and any alternative approach was interpreted as erroneous. This approach completely discourages the desire to study the material and reduces even highly motivated students to zero (Boaler, 2022). It is one of the reasons why they do not want to study mathematics and are afraid of the difficulties that lie ahead of them. Innovative approaches that include constant communication, visualization, and encouragement to find alternative approaches to a problem enhance students’ self-confidence. Thanks to this, it is easier for them to focus on acquiring knowledge and pay attention to those moments that are most difficult.

The book reveals the most important points and highlights the essence of innovative approaches to teaching mathematics. However, after reading it, questions remain that are important for determining the vector of learning. First of all, this is due to the introduction of innovative technologies in the course of the usual process. It is vital to understand the extent to which reformed teaching must supplant the established system. Perhaps, with a sharp shift in the direction of the process, students will not be able to quickly adapt and their academic performance will decline. In this regard, it is necessary to clarify how smooth the implementation and transition to the new method should be.

Another equally important point is the question of how often certain integrations should be introduced in the lesson and whether this will destroy the usual subordination. At school age, students can be extremely violent and not always able to control their emotional spectrum. Teachers can be impacted by these changes when they switch to innovative methods. For example, when introducing a playful approach or discussion debate, students can get overexcited and forget about respect for the teacher. Since the culture of communication with peers is completely different, they often control their speech. Thus, it is primarily a disciplinary question about how one can maintain calm and respect in the classroom while applying creative methods in mathematics.

Despite areas requiring clarification, the author’s research is complete and up to date. Students do experience problems learning math, including fear of constantly memorizing formulas and anxiety of being judged. However, in most cases, students continue to study according to the outdated system, and the author focuses on how one can change the vector of development of the subject. My take on Jo Boaler’s book is that I see it as a solution to a pressing problem. The author offers many ways to simplify and improve the acquisition of knowledge. She gives concrete facts and examples to conclude that her methods are applicable and relevant.

However, there is a gap that requires more clarification and in-depth analysis from the author. Jo Boaler speaks and encourages intuition and freedom of thought through problem solving and learning new material. However, in this case there is a gap between formal mathematics and its application in real life. It is not clear how free students should be, since mathematics is not only a creative approach, but the science of relationships and the accuracy of calculations. The scope of in which cases intuition is a blessing, and in which it requires correction by the teacher is not fully disclosed.

In conclusion, it must be said that the book is a useful and visual aid for educators who teach mathematics to people of all ages. The author reveals many methods that are aimed at improving overall academic performance. Innovative approaches can improve the motivation of students and their desire to gain knowledge through the introduction of communication and game activities. Thus, students are more interested in gaining knowledge and learning about mathematics as a science of relationships.

## Reference

Boaler, J. (2022). *Mathematical mindsets: Unleashing students’ potential through creative mathematics, inspiring messages and innovative teaching *(1^{st} ed.). John Wiley & Sons.