How does anybody define proficiency in mathematics? Various people opine on this question differently. Some say that it is quickness in answering questions; others attribute proficiency to applying strategies with fluency. Well, these are just the outcomes. The actual process leading to perfection involves concept imbibing, practicing, and re-practicing till perfection. Somewhere between the concept introduction and the knowledge attainment lies the process of reflecting. Let’s try to understand how reflective learning helps attain proficiency.

**Taking a pause and reflecting – A lot emerges from it**

We cannot stuff mathematical concepts one after another in our minds. Learning makes sense when we do understand how to relate concepts to one another and apply them to achieve the objectives. Practically speaking, reflecting comes naturally as an intermediate step in the process of attaining proficiency.

Mathematics is a subject where learners move from basic to advanced skills and every basic skill works as a primer to the next. For instance, division is a basic skill, it will be applied to form fractions, percentages and solve decimals too. Reflective learning comes naturally into the process as it highlights the learning gaps and inspires to work harder on those. Here is what reflective learning leads to.

**Offers ease of application**

It all starts with the knowledge of numbers. Kids learn the shape and sound of numbers at the start. Gradually, they learn to compare and then apply operations. Number sense building serves as a template for several math skills and allows math beginners to pick the correct corresponding value.^{[1]} Once it is developed well, the other ensuing operations become easier to master.

Reflective learning helps when you graduate from the basic number sense knowledge gained to applying it in solving math problems. Reflective learning is nothing but going back to the concepts acquired mentally and retrieving them to apply for solving complex problems proficiently; it is quite beautifully explained in the book ‘Adding it Up’^{[2]}. Hence, when applying, you need to reflect on:

- Which strategy to use for solving a problem
- How to quicken the calculation
- How many ways available to solve the problem

With practice, the reflective learners realize the way or strategies they are most comfortable with. And, develop their problem-solving strategy for quicker results.

**Helps understand how far one has come**

Almost every math session starts with a review of concepts taught in the last session. It is nothing but putting the reflective mode on. By asking the questions related to the previous session, teachers stimulate and direct the thought process to revisit the skill acquired the last. Self-learners who have the reflective learning style mentally chalk out the concept development’s progress. They can reflect upon all the concepts acquired by:

- checking proficiency level through tests, quizzes
- solving worksheets; helps find if they have attained fluency in the concept or not
- discussing topics with peers or teachers

When the students reflect upon the concepts acquired, they come to know the level of mathematical reasoning achieved. It helps them ascertain if they need more practice in any of the topics. Automaticity is one of the indicators that determine the level of expertise achieved.

**Helps identify learning gaps**

Reflective learning is all about paying attention to the learning curve. It is not restricted to praising the achievements only. There is a lot of self-evaluation involved using which the learners develop metacognitively about their shortcomings. For example, students may have attained automaticity in adding numbers but when it comes to memorizing the multiplication facts, they fumble a lot.

While solving the multiplication-based problems, the reflective learners can put a red flag on the multiplication facts fluency. When they are not able to solve the problems based on multiplication facts or solve quite slowly due to the lengthier addition way adopted, they realize the importance of mastering the facts.

If sincere in approach, self-learners try to improve by more practice, and not giving up till they feel all facts are firmly fit in their minds. This sense of realizing the need to practice more emerges from reflective learning and is quite crucial for attaining mathematical competence of the next level.

**Allows trying various working methods to solve a problem**

Reflective learners work to acquire a grip on all possible working methods available to solve a math problem. Hence, pausing and reflecting are part of their problem-solving strategy. In one of the studies,[3] the success of reflective learning was assessed for visual and other types of learners. Though present in varying levels, the learners of all styles do use reflective learning in solving problems, despite the fact that the success rate may not always be appreciable.

It is an essential skill that allows kids to adopt a procedure that complements their learning level. Some exam patterns require the students to only pick or write the right answer. In such cases, the preparation phase does include a reflective approach where the student identifies the method most suitable and effective and keeps it in his repertoire of ways to arrive at the solution.

The reflective approach also allows to compare and contrast the time required in every strategy; the frequency of yielding the correct answer is also found out. Thus, the students adopt a procedure that suits them best for their academic performance. They try to keep it in their mind as a learning for a lifetime by practicing it several times.

**Helps check if the previously learned skill is easily retrievable**

Proficiency in mathematics is a function of the smoothness with which a learner retrieves the concept and applies it. Reflective learning involves reading a lot. It is followed by reflecting upon what read. The process also involves solving lots of problems by applying the concepts learned. Since the learners reflect upon the memory recall level, it helps them ascertain if the skill learned has not faded out on the addition of a newer skill in mind.

One of the most important aspects of reflective learning is revisiting past experiences. Since the learners are taking a few steps back and reliving the experience (which formula they applied, where they felt stuck, which parts they found easy, etc.), they develop critical thinking. Consequently, they develop a problem-solving strategy, assess the proficiency level, and build a cognitive understanding of the math skills acquired^{[4]}.

**Summing up,**

Mathematics is a subject that utilizes memory as well as computation skills. Reflective learning allows the students to check their memory’s robustness and the ease they have developed with the problem-solving strategies. This mode of learning is quite useful for mathematical skills development in several ways. The very act of reflecting brings confidence, self-awareness, and the much-needed pause required for processing and ultimate ingraining of mathematical concepts in our minds. Whether it is applicable for all remains a matter of debate, however.

**References:**

- Wagner, D., & Davis, B. (2010). Feeling number: grounding number sense in a sense of quantity.
*Educational Studies in Mathematics*,*74*(1), 39–51. https://doi.org/10.1007/s10649-009-9226-9 - National Research Council, Education, D. O. B. A. S. S. A., Education, C. F., Mathematics Learning Study Committee, Findell, B., Swafford, J., & Kilpatrick, J. (2001).
*Adding It Up: Helping Children Learn Mathematics*(Illustrated ed.). National Academies Press. - Nurul Zahara, Hajidin, & Abidin, Z. (2020). Reflective Thinking Process Students with Visual Learning Style in Mathematical Problem Solving.
*International Journal of Scientific Research and Management*,*8*(09), 187–193. https://doi.org/10.18535/ijsrm/v8i09.m01 - Lee, Shin-Yi. (2021). Revisiting Students’ Reflection in Mathematics Learning: Defining, Facilitating, Analyzing, and Future Directions. Universal Journal of Educational Research. 9. 154-160. 10.13189/ujer.2021.090117.