Mathematics and the Real, Mature World
Mathematics is the key to making intuitive and patient problem solvers. Come join us in making a better world with Math!Ravinder Kumar
When was the last time a kid came up to you and said, ”Another day has passed in my life and I haven’t used the Pythagoras theorem!” The current generation, Gen Z, although is more connected with technology than any of the preceding generations, it surely is facing a stark disconnect to one of the most important fields out there - Mathematics.
Simply put, Mathematics is Logic Reincarnate. As we look deeper and deeper into mathematics and its evolution in terms of pedagogy, we have had a mixed history of success and failure towards making the generations more and more affiliated to Math. In this world of modulated information that is available out there on the internet, it is important that students, and perhaps, teachers (or anyone who has a stake in society and education) should emphasize the need of patient problem solvers. This, as obvious as it may be, is a key link for stronger STEM education out there.
Math - Patience and Intuition
Consider an example where a student is asked to find the amount of time it takes for a water tank to be filled. In theoretical, textbook-based examples, students are fed all the information they need. The dimensions of the tank, the rate of flow, and for higher levels of physical analysis, even the viscosity of the fluid. Conventionally, a student would simply use the formula that she saw in the textbook and then she would apply, give the answer and be proud about knowing some really elementary, reality-disconnected Math. What are we missing on out here? How wrong is this approach?
Let us see step by step what is going on. Usually, students are fed all assumptions, parameters right to them. All conventional systems just require them to quickly solve problems and get a score. This is where we are creating the regrettably huge gap of understanding. Einstein in one of his quotes said:
“The formulation of a problem is often more essential than the solution, which may simply be a matter of mathematical or experimental skill”
There is, no doubt, that rigorous mathematics is needed to have skilled mathematicians who are good at calculations, but the formulation of the problem at hand is, in fact, more necessary than to merely solve it. In our example of the water tank, we understand that the question’s typology is quite conventional and is indeed a concept based problem that needs to be dealt with. Many textbooks have such problems. And as a matter of experience, we all know how to solve it (or at least approach it). But the problem lies in the independence of the students with respect to the problem. They can only apply the formula and get the answer. Instead, they should be able to assume parameters and come up with their own little mathematical models. When everyone creates their own models and solves them to get some answer, and everyone discusses how these answers are relevant and what were the assumptions involved, the teacher can step in and show it to them how the problem can be efficiently modeled, how the solution can correspond to it, and how the theoretical solution can be different from the practical scenario. Since this involves a lot of subjective knowledge build-up, students start experiencing the right kind of intuition towards mathematical parameters and Wunderbar! We have, hence, created patient, intuitive problem solvers.
STEM and Mathematics
Now let us look at another strong connection with the real world and how conventionally math should be helping it. With the advent of the philosophy and pedagogy of STEM, we see a thorough overlapping of subjects. This not only forces students to have an interdisciplinary procedure of thought but also helps them develop critical thinking towards problem-solving. Mathematics is the intersection of all fields in STEM. Its applications in science, engineering, and technology are evident and obvious. However, in a poorly structured curriculum, we see that there are students who can build robots but cannot pass in the subject of electronics. This is not only regrettable but also disheartening to both students and teachers. We need to understand that mathematics should not be the formula to a well-made problem from a textbook, but the vocabulary to solving real-world problems.
Let us return to the tank filling problem. In engineering curricula, especially in fields of electrical and mechanical engineering, students learn a subject of Control Systems. One classic example involves being able to “control” the level of water in the tank. Here, in undergraduate courses, students face immense difficulties due to the lack of intuition in mathematics. Not being able to model a simple water-flow mechanism, students are also expected to control the level of water in the tank. Many undergraduate courses then have to teach how to model these simple mechanisms before even getting into the core of the subject. Not only is this counterproductive but also brings us to question the current state of the elementary and high school levels of education.
STEM and mathematics shall always overlap in a way so profound that even the most skilled engineers, scientists, statisticians, mathematicians, and other professionals can not fathom. The very task of exposing this immense overlap to the generation Z is a daunting task to which we must express concern and give it the right kind of attention.
What can we do?
STEM is indeed the solution to the world’s upcoming problems and we have much to learn from our mistakes in conventional teaching pedagogies. A substantial change on the societal level shall take immense time and make it way more difficult to actually solve problems. However, educators and student communities can do the following to boost the beauty of math in their own schools and environment.
Students can perceive better information with the onset of multimedia in class. In our example of the tank, if students were subjected to a very long tedious video of how long water takes to fill a tank with a rate of 1 milliliter per second, they would roll their eyes, grab their pens and actually find out for themselves the amount of time it takes to fill the tank. Balanced use of multimedia enables students to have better intuition and a clearer perspective without curbing their power to imagine.
Next up, letting students formulate most of the problem is key to understanding the beauty of the application of mathematics. It's not about the solution, as Einstein said. Shorter the question, easier it is for the students to assume, contradict and reason well with the physical situation at hand. Letting them assume parameters, going through a wee bit of error analysis and critical thinking empowers them to make better models.
Only intuition based discussions let students be better involved in class, especially for those students who are not very, “mathy”. The inclusion of all students empowers the discussion and helps them learn how to support their ideas and learn how to logically argue. This, in its own right, is a big step in a world where an argument is slowly becoming the art of offending the rest instead of logically solving the arrangement of the debate or discussion.
Mathematics of a free physical system for a student can be daunting at first, but helping them around a little is required. Do notice the stress on the word “little”. As teachers or students in communities, it is necessary to understand that the appreciation for math comes when you look at a model and perceive the blank paper in front of you as a canvas, and not an answer sheet that shall grade your intelligence. Only then can innovative solutions come up.
In a Nutshell
Mathematics is the vocabulary of the right argument and the spirit of logic. To empower the growth of STEM, we need students to start becoming the patient and intuitive problem solvers. To do so, the use of many techniques is given. However, the key lies in (in fact, has always lied in) perceiving the blank page of a math notebook as a canvas, and not as an answer sheet that shall grade one’s intelligence.