An Absence of History
I recently began a book called An Imaginary Tale  The Story of √−1 by Paul J. Nahin. The book traces how mathematicians regarded negative numbers and their square roots over the past ~2000 years, the birth of complex numbers, the development of complex number theories and their applications in diverse fields. My reading of the first few chapters which paints a fascinating history of the origin of complex numbers planted the seed in my mind from which this post germinates...
When we learned English (or any language) in school, the context in which the story, play or poem was written was essential to understanding its elements  setting, characters, culture, symbolism, humour and irony, objects and props, dress, societal norms etc. For instance, reading Julius Caesar required some understanding of first century BC Roman politics. To appreciate any of the romantic poets (think Wordsworth, Blake and Shelley et al.) required knowledge of the Renaissance and Enlightenment movements that preceded it. In a sense, the history of the text shaped how the text was read.
The Death of Caesar (by Vincenzo Camuccini) 
When we learned geography in school, we covered topics such as rock formation, continents, flora and fauna, layers of the earth and tectonic plates, the environment etc. Geography was one of my favourite subjects  I had an excellent teacher and enjoyed studying how our planet had evolved over many millennia. The presence of species based on available resources and how the toughest ones adapted to survive in harsh climates captivated my interest. Similar to English, the history of the earth and solar system formed a backdrop to understanding geographical facts and phenomena. This history added an ancient charm to the subject that made it richer in my eyes.
Continental drift 
When we learned the physical sciences (physics, chemistry and biology) in school, topics began with stories about the scientists who played a pivotal role in our understanding of the concept. I can still picture my chemistry textbook taking me from Dobereiner's Law of Triads to Newlands' Law of Octaves to Mendeleev's early version of the periodic table to the modern version built by Moseley et al. How each version aimed to modify and build on the previous one by incorporating new discoveries on the nature of matter greatly appealed to me!
In physics, the tale (whether fiction or fact!) of Archimedes shouting "Eureka" on realising that the volume of water displaced when he got into his bathtub must be equal to the volume of his body that he had immersed was a comical one used to introduce how to find the volume of irregularly shaped objects. In fact, the story goes that Archimedes was so keen to share his discovery that he jumped out of his bathtub and ran naked through the streets of Syracuse! Similar interesting tales involving Newton (gravity and forces) and Galvani (notion of animal electricity that lead to the discovery of current electricity) served as hooks to these concepts.
In biology (another one of my favourite subjects), examples such as Hooke's observation of cells in cork and Fleming's accidental discovery of bacteria not growing around a mould in his sample of Staphylococcus bacteria (leading to the discovery of penicillin that ushered in antibiotics) were a part of information boxes scattered through my textbook. Theories on evolution and studying how animals and plants adapted over time to their environment was integral to understanding how living things came to attain their current form.
Robert Hooke's microscope and slide of cork cells 
Drawing attention to the fact that theories and discoveries didn't magically appear out of nowhere but were the work of scientists lent a depth to these physical sciences. This added to their relevance (in my mind, as a child) as bodies of knowledge accumulated and understood by humans over centuries.
Nahin's colourful picture of how roots of negative numbers flummoxed many a mathematician until a few geniuses hit upon a solution made me look at complex numbers in an entirely new light. It made me think about how history played some role in all my school subjects barring mathematics. This brings me to my wondering... why don't math textbooks and curricula present and connect how a topic/branch of math was born with the main concepts being learned in the topic? Why does school math basically come down to children recalling formulae and performing numerical calculations under the time crunch of a test?
It is not that educators are unaware of this issue. The National Council of Educational Research and Training (NCERT) in India publishes textbooks for the Central Board of Secondary Education (CBSE)  one of the popular education boards in the country. The NCERT blueprints are adopted by many state government education boards while drafting their own textbooks. A few chapters in math in the NCERT books do have information about mathematicians and history (see the examples below) but these tend to be glossed over as mere footnotes, are definitely not part of any assessment and are no longer referred to once it is time to solve the exercise problems at the back of the book.
So, here is my hypothesis:
IF
we (educators) can show children, in an engaging manner, the
value of a math topic and how it was born out of a need to solve a hitherto
problem or puzzle and use that as both a hook to start the topic and as a
reference during the process of learning the topic,
THEN
children will begin to appreciate the wider scope of math and
stop viewing it purely in terms of formulae and calculations.

Of course, such an idea will fall flat if tried either halfheartedly or in isolation. Changes in textbook design, classroom teaching and assessments would have to take place in parallel to give this idea a fair chance...
What do you, the reader, think? Do you see any merit to this hypothesis? Is this an avenue worth testing out?
~ o ~ x ~ o ~
(A concluding note: The ideas expressed in this post are largely based on my experiences as a student and teacher in urban schools in India. Furthermore, it is grounded in my opinions and vision for how mathematics education can be improved. There is no one silver bullet that can enhance math education and it will require a multipronged approach; what I am proposing here is a suggestion for one of the prongs.)
Hello bhaiya it took long for this post. I wanted to tell that this post was full of your interesting subject and factual data. In the end you said we must design the education system in such way that students must be engrossed in studying. According to me the is demonstrative teaching and using technologies such as projectors,ppts etc. Also including story type teaching for subject like History,Geography.
ReplyDeleteWe pupils look things which fascinates us. So today's generation love technology so studying student behavior and teaching will be very interesting
It's something I believe is super important. I think post high school there's a very stark division between who likes maths and who doesn't. I believe that a lack of narrative makse math a little dry and unrelatable.
ReplyDeleteI think that this is a very genuine idea and worthy of exploration. I too felt similarly when I was teaching at AVBS and tried to do the same with number theory. While I intoducing number systems ( _because, well I had to start with that_)
ReplyDeleteI feel that it is something which complements the idea of constructivism and building knowledge in a classroom rather than just accepting it as it is. Rather than looking at it as something immovable and immutable, looking at it as something which was born out of ingenuity. Such an idea, i feel, let's students look at mathematics as really the _tool_ that it is. Rather than a set of commandments which fell from heaven
While reviewing a paper, one of the things editors look for is if the study is "well motivated" in the manuscript  whether the authors convince the reader that the problem is a worthwhile one to solve. What you've written about feels like the school equivalent of it: kids relate to material if they are given some background and know why they're being taught what they're being taught.
ReplyDeleteThe post actually reminds me of physics core courses as taught at BITS in 2011. We had mathematical methods in physics(MMP) which was taught in sem 1, along with quantum mech and other courses. MMP felt extremely dry because none of the concepts were taught with any background(lets set aside my ignorance for now). For instance, I didn't know why we were learning Laguerre polynomials, till it came up in the solution of the Schrodingers equation for hydrogen atom in quantum later during the sem. Then it all made sense.
In sem 2, for stat mech, many in the batch ended up learning MMP pretty much from scratch, because at that point we needed it. Only after the introduction of the problemdid we became interested in the way solve it.
Showing the value of a math topic and how it was born out of a need to solve a problem, is relatable and I feel the problem should always be introduced before the concept. Without the problem all math is abstract. With one, everything has meaning.
I've been reading up a lot recently about the importance of a "hook" to every math lesson. Zaretta Hammond wrote a great book about Culturally Responsive Teaching and the Brain, and how without these hooks, students don't get the dopamine hit they need to start engaging with the topic. It's interesting to see how and when history connects with students or when it might miss them. When I mentioned pythagoras and the fascination / revulsion around irrational numbers, one student remarked gleefully "I'm greek!" and the historical hook clearly had a positive impact. It can be challenging to find equal representation, or historical figures and contexts that resonate with students typically underrepresented in math. What other ways can hooks of history make the context and the stories of the individuals discovering the math engage student learning?
ReplyDeleteHi Shreyas,
ReplyDeleteThis is an interesting post. Totally in agreement that background knowledge of a place, people, times, environment is necessary. And bringing this view point in mathematics will benefit. Maths is a very interesting subject but unfortunately lot of students are apprehensive about the subject. Wish the syllabus authorities realise this soon and make amendments
That sort of thinking carries over also, both in students and teachers I think, even after school. Even in your MBA course also, people especially engineers, pay attention to the formula that the prof shows, which comes on the 5th slide or so. Everything before and after that they ignore as 'faff' and many times, in case studies, focus so much on searching for the right formula to "apply" to the case that they don't care about what the problem is, why they're solving it, whether that formula or model they've used achieves it, recommendations etc.
ReplyDelete