Alice in wrong land!

- Yathirajsharma M. V.

Introduction to the article: This article presents a viewpoint where the author expresses regrets regarding the systemic failures in achieving educational goals, using an example from undergraduate mathematics classrooms. The opinion is not general in the sense it addresses all the targeted audience but general in the sense that it addresses most of them.

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Alice was once told to find a cat which was born to a dog. Alice ran in search of it!

My journey began in a Non-Government Organization (NGO) dedicated to support public education system. Rather than working directly with students, our focus was on supporting government school teachers in their professional growth. Regular visits to schools allowed me to interact with children, giving me valuable insights into teaching methods (pedagogy). I often found myself amazed by the different ways children thought and wondered why they responded the way they did. Personally, I was curious to know how they understood mathematical concepts and how they perceived mathematical objects. It became clear that there was a problem with how math was taught. Children were taught algorithms without emphasizing the logic behind them. For example, they were taught to add numbers using carrying over without the idea of why. Additionally, children were getting tired before they even got to real-life math problems. This tiredness made it difficult for them to apply algorithms correctly, i.e. to choose a proper operation for the given scenario, even though they had a correct intution of what could have been done. The focus on getting a single-line answer, like for the sum of 18 and 24, showed a lack of emphasis on true mathematical understanding.

Let me share a situation that really made me wonder about the skills that are being developed in children. Perhaps many of you have encountered something similar. When children were asked a basic math problem like 28+32, surprisingly, many answered 5 10. You may guess how they got there. They added 2 and 8 to get 10 and 2 and 3 to get 5. They knew which numbers to add, but they missed the concept of carrying over. Even when corrected, they didn't fully grasp their mistake. To them, both answers seemed right; the second one was simply what the teacher expected.

This situation makes us seriously question the skills being taught. Imagine someone getting punished without knowing what they did wrong. It's similar here. If someone can't realize their mistake, what's the point of education? Has the meaning of 'educated' changed? This issue isn't just in schools; it's widespread. Students are made to keep exam scores as their objective, not their ability to think. Are we, the teachers, rushing to complete syllabus keeping examination as focus? Are the notes of students filled with stars, double stars, tripple stars indicting the proability of appearence of the conecpt or problem in examination? Because results matter for admissions and accreditation, are teachers and institutions prioritizing reinforcing the ignorance over educating them? The ASER report shows the same situation (although I personally do not like what is being tested, I'm just pointing out the data so that we know students are really struggling).

Let me explain how even in undergraduate classrooms, we see the same problems. There are many examples to show how the system isn't working well. There are plenty of examples to illustrate this. I'll choose one example to explain a well-known algorithm taught in complex analysis. This example also works well to show a situation similar to carrying over numbers when adding.

Recently, while interacting with some of the undergraduate students during a viva-voce on one of the nice algorithms of complex analysis namely Milne Thomson (MT) algorithm for the construction of analytic functions over given domains, I gave them a problem to solve. It was this

Find the analyitic function whose real part is \(\dfrac{x^2+y^2}{2}\).

Those who are not aware of analytic functions click here. Few of the students agreed that they couldn't recall the algorithm. But those who knew it were very enthusiastic in applying Milne Thompson Technique and finally bursted out in exaggeration that it was \(\dfrac{z^2}{2}+c\). I asked them if they were sure that the answer they got was correct. They confidently showed me the method they employed and asserted that they had done in the same way they had been taught the algorithm. Yes! They were right in the way they had applied MT algorithm (the way they were taught). It was this they had done:

Step 1: Take \(f(z) = u+iv\) as the required analytic function. Given that \(u=\dfrac{x^2+y^2}{2}\). We have \(f^{\prime}(z) = u_x+iv_x\). By Cauchy-Rimeann Equations: \(v_x=-u_y\). Hence, \(\boxed{f^{\prime}(z) = u_x-iu_y} = x - iy\).

Step 2: Setting \(x=z\) and \(y=0\) we would get \(f^{\prime}(z)=z\).

Step 3: From above, one would get \(f(z) = \dfrac{z^2}{2}+c\).

I asked them to find the real part of their final answer and also asked them if there was any condition on the constant. For them \(c\) was a complex constant. They verified, and got that

\(\Re\left(\dfrac{z^2}{2}\right)+\Re(c) = \dfrac{x^2-y^2}{2}+\Re(c)\).

They were unable to understand my question. I once again asked them, if their asnwer was correct as they were not getting the real part \(\dfrac{x^2+y^2}{2}\) from their final answer. This again remainded me of my good old days of interaction with children in my previous work: "\(28+22 = 4 \,\, 10\)". They weren't ready to agree that their answer was wrong and a slight change in this scenario was that, they felt that \(\dfrac{x^2-y^2}{2}+\Re(c)\) might yiled \(\dfrac{x^2+y^2}{2}\). That is, they were thinking that what they did was also right without denying that the other answer was also correct.

Alice was once told by someone to find a cat which was born to a dog and alice went in seacrh of it! Students couldn't realize that such an analytic function whose real part is \(\dfrac{x^2+y^2}{2}\) could never exist even when their answer was hinting at the same. This probably lies in the fact that nobody ever discussed such an inexistential problem with them. All that was discussed with them was how to answer a question which appeared in the form 'Find an analytic function whose real part is so and so' in their question paper.

If we take a moment to think about it, there are two things that might concern us. First, many of us believe that people don't enjoy thinking about things they understand. Instead, we think of thinking as something tiresome. Second, we often avoid thinking ourselves because we haven't been encouraged to appreciate it, maybe due to how we were taught without enthusiasm. This cycle continues. But the ability to recognize the possible limitations of a process, the skill of interpreting the final output or answer, the skill of rectifying the mistakes in one's own arguments are more important skills than just reprodusing the procedures or facts leanerd. I think no one can disagree with that.

What could have been done? Once a procedure or an algorithm is taught, one should also discuss the limitations of it, if exist. It would also be better to discuss the logical reasoning behind the procedure if it is within the reach of the existing conceptual knowledge of students. I do not deny with the fact that not all students would catch what is beng discussed. However, I believe that lack of such discussions would really seem to be simply eating something just because one has to survive! The beauty of some algorithms is that they would give answers even if they are applied on wrong candidates. Milne Thomson is one such algorithm. For applying it, the candidate that had to be chosen should have been a real part or imaginary part of some analytic function. This should be known in advance. But, even if some function \(u\) is not a real part or imaginary part of some analytic function, application of Milne Thomsosn would yield an answer, as in the example discussed in this article. What is the use of spending our energy on a wrong candidate? We should have had spent our energy on identifying that the candidate was the wrog one.

By the way, it is quite easy to see that \(u=\dfrac{x^2+y^2}{2}\) cannot be a real part of any analytic function as \(u\) is not harmonic, that is \(\dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}\neq 0\). We know that both real anad imaginary parts of an analytic function are always harmonic.

The original paper of Milne-Thomson, which can be found here, couldn't be followed so easily by me as it involved very vauge procedures. But one thing was sure for me then that Milne Thomson was studying the function by reducing it onto the real axis. I thought that Milne-Thomson had made many assumptions and to arrive at his technique, I sought a different kind of argument using some sophisticated tool of complex analysis, so called 'Identity Theorem'. What I had sought ran exactly on the similar lines as it has been given in this page (answer of Christian Blatter) except that I had made few assumptions that the first partial derivatives of \(u\) looked like restrictions of some analytic function to real axis after the substitution of \(x=z\) and \(y=0\).

Considering the background of our students, I agree that explaining the rationale behind the MT algorithm could be challenging. However, Milne-Thomson's intuitive idea itself is intriguing and might leave people in uncertainity. The bottom line is: there is a dire necessity to move beyond just exam preparation and prioritize developing various skills that these lessons are actually meant to teach us.

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Yathirajsharma M. V. currently works as an Assistant Professor of Mathematics at Sarada Vilas College.

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