An essential part of learning mathematics is about reasoning and making sense. What does this exactly mean?
When a student is given a problem, he needs to make sense of it, from his level of perceptive which is unique to each individual. This will come with big struggle, and the important next step is to stay motivated, curious, be extremely perseverant and not give up after the first few attempts. This might also require a good relationship with mistakes.
A students will have to develop his own strategy to solve a given problem. That might imply first to translate it in their own language, use their own words and knowledge background to get (understand) the actual question and problem they are attempting to solve.
They will have to build bridge in their mind to similar problem they have solve in the past even though they might seem different. This bridge will be easier and easier to connect with practice and experience and sometimes might not work and some other connections will need to be created until finding a suitable one. This can be interpreted as finding pattern. Critical thinking is a lot about those pattern, finding pattern is to me a form of the ability to abstracting. One would have to look for and express regularity in the whole of the experience with mathematics. This sometimes happens subconsciously.
It will be also important to identify the information necessary to answer the problem. When a plan has been identified and tried and an answer have been obtained, it is always important to have a critic look into our work and insure we are making sense. In the sense that someone else will need to understand us this would mean to be able to detach ourselves from our point of reference and being able to adopt an external point of reference making sure all the step needed for most to follow are there. In my experience, this step is often missed, understated and found completely irrelevant. Making sense in english, no need of sophisticated perfect english is I believe essential. I often recommend to read the sentence you write in your head in english (or translate into the language you feel most confortable) making sure at least it structurally and logically makes sense, often the problem already lies at this level and there is a belief that there is no need to make sense in english (or the language used to write mathematics). Even mathematical sentence should be translated into our most used language into our mind to try to insure that. Mathematics is a language and translation into a language we are used to helps to make sure the essential part of making sense in the end is achieved.
Mastering the art of sharing knowledge and being understood by others is undoubtedly essential in any day to day life to socialise and work efficiently with others.
Another important step, would be to evaluate if there is another plan that permit to get the answer compare it the previous one and understand the benefit of each. In addition to that, it is also always nice to have strategies to double answer, it could be other methods, or just making sure that the answer given is not a nonsense in the context of the question or impossible. Like a negative distance or a negative area etc… Often one can identify strategies like this and this help with being sensible and critical thinking.
Making sense and reasoning is a process with subconscious and conscious phases. In my opinion, if we become aware, in other words more conscious about our own steps in this process we become more efficient.
This process involves necessarily different forms of abstractions, abstract reasoning. Abstraction that permits to compare old techniques and problems with the new one, abstraction to understand the substance, the essence of the notions involved. That might translate into an idea of modelling with mathematics identifying the appropriate tools strategically. But also, more quantitative reasoning, that might involved numbers that might be perceive more concrete as the students might have become used and more comfortable with them.
It essential to being able to create strong and valid argument as objectively as possible. Strong and valid argument are often identified by a logical and meaningful structure. It might be essential for that to collaborate: receiving critics from others and discussing them, filling the gaps and fixing mistakes is essential. But, it is still important to understand others point of view and being able to compare do valuable and helpful critics to improve if necessary or understand better others.
I extracted other ideas around reasoning and making sense from the paper: “Mathematical Reasoning
and Sense Making” Michael T. Battista.
“Reasoning and sense making are critical in mathematics learning because students who genuinely make sense of mathematical ideas can apply them in problem solving and unfamiliar situations and can use them as a foundation for future learning. Even with mathematical skills, “[i]n order to learn skills so that they are remembered, can be applied when they are needed, and can be adjusted to solve new problems, they must be learned with understanding [i.e., they must make sense]”
(Hiebert et al. 1997, p. 6).”
“Sense making is also important because it is an intellectually satisfying experience, and not making sense is frustrating (Hiebert et al. 1997). Students who achieve genuine understanding and sense making of mathematics are likely to stay engaged in learning it. Students who fail to understand and make sense of
mathematical ideas and instead resort to rote learning will eventually experience continued failure and withdraw from mathematics learning. ”
“To genuinely understand mathematical ideas, students must construct these ideas for themselves as they intentionally try to make sense of situations; their success in constructing the meaning of new mathematical ideas is determined by their preexisting knowledge and types of reasoning and by their commitment to making personal sense of those ideas.”
To conclude, an advice from me to improve that skill is to question yourself when you are saying something even in your day to day life: “Is this exactly what you meant even in your usual language?”. You might need to ask others what they understood from you and compare the three: what you wanted to say, what you said and what was understood and evaluate objectively the reason of the gaps between them if any and improve when necessary. If the way you think you communicate does not correspond to the way you are perceived it might mean that there is a missing harmony between what you want to say, communicate, appear and what you say, act and other perceive. Because in being precise and making full sense you are already doing mathematics (a bit of logic, reasoning, structuring argument, critical thinking) to me. It is your choice not to care about making sense but a lot of time what you say does not match what you thought of saying, because it has become so clear in your head that it is difficult to understand why this could be confusing or not enough to make yourself understood in the first place. This might get very frustrating and make work less efficient. Mathematics are useful in making you communicate better and improve your arguments and sharing meaningfully your experience.