Rethinking the way we teach children about numbers
Calling all teachers of Mathematics to stop a moment and think - are we really teaching it right?
One of the joys of being in Education is the implicit permission to enjoy the teaching-learning process. And one of the great satisfactions of being in this field, is the permission to reflect on how to teach it the best. Occasionally, this process of reflection leads to a frustration with the status quo ("Arrgh! They are not teaching it right at all!") and the deep wish to somehow become Elon Musk and start one's own school! :D Of course, most can't really do that, so we write instead on a public forum of thinking people, hoping to get the right people thinking about it.
The way we teach Mathematics (almost everywhere) isn't very good in building the right sort of intuitions about numbers, number patterns and in relating to numbers at scale. This occurs to me every time I work on a problem relating to infinite series. If Ramanujan concluded with no consternation that 1+2+3+4+... = -(1/12), it was because he had an unburdened mind around how to think of numbers. He was free of the shackles of orthodox ideas and imposed restrictions. He had a truly free mind to learn it right, and to imagine.
Most of us will, though, will scoff at this result and declare it dead-on-arrival by discarding it as 'absurd'. Only thing - it is not. Srinivasa Ramanujan's notes contain references to this amusing result. Those who know the Reimann Zeta Function know the full story. Take it from me - it is valid (or look up the internet). The proof for this is surprisingly intuitive, if we let go the stranglehold of our traditionally taught notions of how numbers work. Ramanujan appreciated it immediately. But it eludes us in today's system. Why?
The problem is two parts - 1) we teach from a limited framework and 2) we are too categorical in what we say.
We declare categorically that addition is commutative and associative and that identity property holds and so on, without as much as leaving a snatch open for a possibility otherwise. We teach that numbers increase perpetually on the positive side of the number line; and decrease perpetually on the negative side. We offer some convincing enough proofs to make our case stridently too.
We have shut the door to curiosity, and to contrarian possibilities. Do numbers _always_ work this way? All numbers? All the time? Are there places where this may not be true? Of course, for a 6 year old learning arithmetic properties for the first time, such contemplation is not in order, and neither am I suggesting that we make the Mathematics curriculum any harder than it already is! But what stops us from stopping around grade 7 and thinking - what happens to numbers as they become bigger and bigger and bigger? What stops us from introducing infinities to a high-schooler? The fun in Mathematics really is the pull that it exerts from the far recesses of the unknown, beckoning the reader to reach for it - the eternally playful, elusive, but bewitchingly alluring tease. With everything being 'informed' to students in categorical, unquestionable ways, the element of teasing and alluring is altogether lost. That is a shame. Curiosity does so much more for sustained interest than the pursuit of marks in exams ever will!
In my mind, the attraction in Mathematics is from the possibilities and unknowns it holds, and at the same time being definitive about what it knows and states to be true. This is curiosity-evoking, but then, the dichotomy often results in misunderstandings in going from the definite to the unclear. For instance, many teachers understand infinity to be a very very very large number. If a student learns this and believes this to be the final truth, a lot of higher Mathematics is going to be seriously elusive. Any wonder that they struggle terribly in later years, as large parts of Engineering, Science, and most certainly Mathematics, become incomprehensible or plain absurd?
If infinity is poorly understood, its corollary, the infinitesimal will also be poorly understood. And that's the reason so many students simply can't wrap their heads around ordinary ideas like the probability density function. "Why can't I read a point-probability off a PDF function?" They struggle to understand where it is OK to approximate and where it simply won't hold. In the case of infinities, disabusing the learner of the idea that infinity is simply a very large number, is a fundamental correction that needs to be brought to their thinking. If they don't understand that infinity is an abstract notion of the endless, and that it is easier thought of in terms of set theory (into/onto/bijections) or probabilities, than in terms of ordinary properties of discrete addition and subtraction, they will perpetually struggle with how diverging summations converge at infinity.
Do students understand that numbers oscillate? Actual, tangible numbers! Most don't. And that is why the idea that a result like 1-1+1-1+1-1... converges is so hard to comprehend. We have taught them wrong. We taught an over-simplistic idea of numbers and never bothered expanding their field of vision while in school. Again, the idea is not to further expand the Mathematics syllabus by introducing more content and rigour. No! The idea is to bring in wonder and the ability to deal with the abstract, and to introduce broader ideas earlier, even if gently.
Let me give you an example.
The notion of analytic continuation is really not that hard, in itself. It is just a nice, inclusive world where the so-called "imaginary" numbers are also made to feel less esoteric and more at-home! It extends the domain to include complex numbers. Once you get your head around how complex numbers influence real numbers, and how the number line is far from a straight never ending perpetually increasing (or decreasing) line, you will start to see how partial sums can bend! The number space of all numbers is a fascinating place. A truly fascinating place where perfectly reasonable results happen which otherwise, from a more limited "straight number line" world view does not permit. Helping students view numbers along more than a "line" and helping them see the way number components get rotated in the number space, will lend a lot firmer foundational framework to work with even high-school level concepts.
Let us put it simply and start with the basics - the words we use to convey ideas. If complex numbers are taught as real and imaginary, students will perpetually stay befuddled about how imaginary things can create effects in the real, physical world! Gauss himself preferred that they be called lateral rather than imaginary! Call them lateral; call them rotational - suddenly, the physical impact of complex numbers becomes umm... far less complex! They become relatable. Today, by calling it imaginary, the whole idea of the rotational effect of 'i' is lost on learners - and that's such a shame, for it is such an elegant, beautiful and fundamental idea!
My tirade essentially is, in summary, only a simple entreaty: Allow students to see numbers as more than just letters to be manipulated. Let them see the visualization; let them imagine the bigger space; let them be gently ushered into the fullness of ideas. Allow the introduction of broader ideas; and let us let students imagine, question, make mistakes, and truly appreciate the broader greater universe for what it really is. Open the windows. Let the possibilities come in.