A Possible Way of Teaching Mathematics!

Mathematics is very intricate topic. It is hard. Sometimes, the ideas can be understood with less struggle, but to pen down with great technicality is really hard. Writing down what you are thinking, imagining inside your head with using just words, is respectively less hard. But, writing down those ideas, thoughts and imaginations with mathematically rigorous language is really really hard. No matter how much one knows math, penning down is difficult. To make it less difficult and to understand it locally and globally, one learns as much as possible. There is no supremum.

It is very reasonable to think & state: “Do not try to make math as easier as possible; rather try to make it as less difficult & more understandable as possible.”- The Author

Purpose of This Article: This article will deal with the possible wrong approach of teaching math to identify the possible effective one. What does make it wrong and unhealthy? Why is it effective and why is it needed? What is the possible solution?

Audience Type: The Math Community. The people who are teaching, no matter what the subject is.

Take Away: A possible solution to A Wrong Approach of Teaching Math . The Glorious Day: Writing down the imaginations mathematically. The Glorious Day is extremely hard to reach and very rare. To measure the difficulty, Mathematical Plates has been included.

Mathematical Plates has also been included for fun.

Implementation: This solution can be implemented in any subject to some extent.

Mathematical Plates: The following mathematical plates have been added so that one can understand that these objects have some layman interpretations as well as deep mathematical rigorous interpretations. The latter one is much more difficult.

Plate 1. Triple Torus. How can it be explained mathematically? Background Needed: Algebraic Topology.
Plate 2. K-3 Surface. Where does this come from? Background Needed: Algebraic Geometry.
Plate 3. Conway’s Topograph. Picture Source: Web Search. What can you think after seeing this? Background Needed: Quadratic Form, Graph Theory, Hyperbolic Geometry, Complex Analysis.
Plate 4. Apollonian Gasket with Farey Resonance Lines. It can be derived from Conway’s Topograph. How can you relate it to Plate 3.? Background Needed: Same as Plate 3..

The Possible Things, Going Wrong in Teaching Math:

  1. Those who do not want to teach but research, are teaching.
  2. Rushing for the quantity and syllabus only.
  3. Quality and quantity with respect to a standard mathematician but highly technical with respect to a student or a newbie.
  4. Recommending books by big-shots unnecessarily.
  5. Recommending same books for all.
  6. Less visual interpretation.
  7. Sometimes, good and easy sources are not being used.
  8. Sometimes, giving long Theorem-Proof.
  9. Less examples and applications.
  10. Sometimes, giving low marks. Below passing marks.
  11. Not using online resources.
  12. Giving exceptionally difficult problems in a short-time exam and too many of them.
  13. Same approach for every subject and every topic.
  14. Ineffective marks distribution.
  15. Sometimes, giving short time to think.
  16. Usually one professor/teacher is given a topic to teach.
  17. Taking exams every week or so.

The Possible Solution: Broadly, the idea of possible solution is Classwork at Home & Homework at Class. Some of the ideas, Mathematical Plates and Examples are heavily influenced by Dr. Vineeth Chintala.

  1. First two weeks, only main concepts, ideas, visual interpretation. No technical things.
  2. There are plenty of good YouTube Lecture Series for basic courses. Use it as basic standard material if it’s good. Take Less classes; may be twice or thrice a week to focus on problems, examples and extra concepts only. Create groups of 4/5 people for problem solving where each group will give presentation of another group’s solutions. Plan it beforehand. Be generous of giving marks only for the presentation. (Maximum marks is 15). 3 Group Assignments will be there of 3*25=75 marks and each 25. Keep Second Assignment relatively tough. Only marks of two assignments will be considered; that is best 2 out of 3. Then, rest 35 is exam based. Then, the marks distribution is: 50+15+35=100. Self-Learning issue for some students will not be there because there are very good lecture series available on YouTube.
  3. The Complementary One: A class room environment can be created such that YouTube Lectures Series will be shown on regular class timing in stead of Physical Professors’ Lectures. Make it optional. Some students can learn from home/room or come to class. This point has been added because some students might feel like “Not Learning anything staying at home/room or having some issues including internet”. Occasionally, these classes can be monitored by PhD Folks & PDFs.
  4. A Glimpse to Rare Reality: Project Laboratory in Mathematics: A Taste of Research.
  5. Take 2 lectures on every one month interval to know if everything is alright and what is the need & want.
  6. Math is hard as well as life. Give enough time to think & compute.
  7. First give intuition and then possible technical terms whenever necessary.
  8. Teach them to draw diagrams and flow-chart of the learned topic up to independence.
  9. In addition to 2, give references of Intermediate-Hard-Advance books.
  10. At least two instructors/ professors/ teachers.
  11. No frequent exams. Be flexible.
  12. Build foundation slowly. Give time to think & grasp. Then, gather momentum.
  13. Printed Notes must be available for Xerox (If and only if Note is required).
  14. Every subject should be taught in a comfortable manner. For example: Graph Theory can be started by solving some interesting problems without too much technicality. Group Theory can be started with developing algorithms for solving a Rubik’s Cube or how much one can modify a set by using some extra structures rather than telling the direct definition of Group. Connect bridges between two subjects without so much technicality for that moment.
  15. Set a Separate Class “Sketching up the right things from the wrong ones”: Where a student has to write a wrong proof and another student has to correct it and modify it. Here, wrong means conceptual and conceptual-computational wrong. Not Unnecessary Computational Error. There will be a third student whose work will be only asking questions about that right proof. A “Trilogy” will be formed.
  16. At the end of each month, chocolate distribution might be a good idea. For everyone.
  17. Tell them to have a full stomach & sound sleep. If someone hasn’t slept well last night like me (long back) and caught tired & yawning(long back), tell him/her to sleep in the class (last 2/3 benches) or room(according to his/her preference).
  18. Tell them to spend some time doing other things whenever enjoyable.
  19. Tell them to start “Little Math Magazine”, “Weekly Student Seminar” for the students, by the students, of the students.
  20. Tell them about Time Management & Life Hacks. Example: There are two books- A(Hard; average reading time: t), B(Intermediate; average reading time: s). Sometimes Counterintuitively: t ≥ s (First Reading B) + t (After Reading B, Read A).
  21. Tell them to contact senior folks.

Pros With Respect to Students:

  1. Less pressure. Enough time. No huge tension for passing marks. Fun.
  2. Chance to lean and understand math concepts. Quality & Quantity.
  3. Engagement, Discussion, Gaining Confidence.
  4. Multicultural backgrounds.

Cons With Respect to Students:

  1. Too many things are easily available.
  2. Not worrying for the passing marks.
  3. Too much marks of too many students. Hence, a problem occurs in categorizing & ranking the students.

Pros With Respect to Professors:

  1. Can do teaching and research at the same time. Less pressure. Less teaching. Will have time to do other things. Less Answer Papers Checking.
  2. Cool Math Professor!

Cons With Respect to Professors:

  1. Too much flexibility.
  2. Making some math notes and TEXing it.

Time For The Professors Has Come: Those aforementioned 21 solutions are given and structured in such a way that if one student exploits the flexibility and independence given by the professor, s/he will be spotted without making any effort. Therefore, this can be dealt with Solution 2, 13, 14, 15. Hence, Cons can be reduced to some extent.

Post Scriptum: While, writing this article, I thought to include “To let anyone eat food in class if s/he is hungry” in the solution. But, then it would be too difficult to handle. It is just because of the distraction of that delicious piece. I would have taken a piece or two rather than watching him/her or doing math!

“Math is hard. So is Life. Keep Math-Life Balance in sync and keep the jokes alive!”- The Author

Hence, it is complete.

Cheers! Picture Credit: Google Search

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Math Researcher. Writer. Professional Athlete.