Lenses for Numeracy
While this post has to taken in the context of the new initiative of designing a curriculum (which fosters development of qualities versus content recall and application), an intelligent reader will find multiple ways of extending what s/he reads below into their particular setting. The earlier posts in this thread include ones on the history, philosophy & nature of qualities that we wish to develop at Purkal Youth Development Society (PYDS).
This particular post focuses on a technique being explored with the team on how to take mathematics out of the lesson and into life. This, I strongly posit, is vital in releasing any fear or discomfort that students develop regarding Mathematics. I must warn the excited reader that this will be a long journey as the teachers you might be coaching have (most likely) never looked at the world like this. They have (most likely) never taught mathematics like this. You would need to be patient and take them on this journey. It is worth the mind-wrangling.
This is a vital change of perspective towards numeracy. We have often been told that here is a language (and mathematics is a language and, at its height, a philosophy) that you must learn because you might need it. It is like teaching students Klingon with the promise that we might need it when invaded by aliens. Well, not that extreme, but you get the picture. Detractors (like the ones who coyly made the illustration alongside) are quick to admonish the language for being irrelevant and arcane (when using kind words) and something that will never be used. The most commonly bashed topics are calculus, complex numbers & trigonometry — who will ever use them!? The primary problem is that we do not extract mathematics from the world around us and instead keep nurturing and widening the chasm between our real world and the language and application of mathematics. The approach outlined below, seeks to address this disservice done to the study of numeracy and mathematics (largely focusing on arithmetic).
While the focus of the new curriculum will remain on the 9 qualities that need to be developed, we are also cognisant of and committed to the needs of entering into grade 4. One of the key skills that a student entering into grade 4 needs is that of performing operations and manipulating numbers. While grades 1–3 will be developing these in the context of the themes and adopting a range of innovative and creative means, the goal of readiness for grade 4 is not to be treated lightly.
Understanding the Lenses Better
Each of the lenses shown above help the teacher understand how to curate experiences for the child so that s/he may be able to identify and understand the play of numbers in whatever they see around them. One big handicap that we acknowledge and address by the technique above, is to rid the child of a thought process that Mathematics and Numeracy is only something done on paper and are essentially a bunch of problems which only a few can solve. The rest basically must grind their teeth till 10th grade before they can opt out of Mathematics! This is a very toxic culture that schooling & common teaching methods, foster. We want the child to be able to look at shapes everywhere & patterns in every action and thing as well as a play of numbers around them. Giving them a confidence & familiarity with how countable entities (vs numerals) behave in the real world is the first step towards making their numeracy skills well rooted.
We always begin with the theme of the month. Themes in our school include topics like “Our Ecosystem”, “Village Administration”, “PYDS and my school”, “Common Tools/Machines we use” etc. In later posts, I might elaborate on each Lens in the context of a variety of themes. This might help make it more concrete for the reader. One more thing deserves mention — We don’t teach one “Mathematics chapter” at a time. We explore a theme per month. What this means is that we do multiple chapters per theme. We would be exploring shapes, counting, operations, currency, time, etc. in a theme. We will reapply them in the next theme and so forth. This has many advantages:
- Students are immediately able to see how each theme is lush with numeracy and lends itself to the language of Mathematics
- Each “chapter” is revisited multiple times in multiple contexts. This trumps revision any day.
- With multiple themes revealing nuanced versions of “mathematical operations” the language of mathematics gets richer in the young minds.
Lens 1 — In Real Life:
Given a particular theme, what are the common actions, behaviours in the context of that theme which deal with patterns, shapes & countable entities? This is the question that we answer under this lens. Some teachers consider/prefer to apply Lens 2 before Lens 1. Some might prefer iterating between the two. The value of this lens is to simply bring to the surface all the common actions & behaviours while exploring the theme and look for shapes, patterns and numerical interactions. My reservation against doing Lens 2 first is that once we list the countable entities, we will be blinded to other actions and behaviours in this theme. E.g. if the theme is Village Administration, then the countable entities include the houses, roads, Panchayat members, electorate, grants, etc. If I, now, return to Lens 1, I am subconsciously ignoring the real life action of the Sarpanch listening to grievances because there is nothing countable about that. But had I started with Lens 1, I might have included “Sarpanch addressing grievances” to then look at the hours that a Sarpanch works as a countable entity & hours spent addressing grievances as a fraction of that (and compute percentages, etc.). It is ok if an “In Real Life” action doesn’t seem to yield a numerical experience; move such an action to the bottom of your list of Lens 1. Tomorrow might make you wiser and you might return to this list.
Lens 2 — Countable Entities:
This seems like a fairly simple and obvious lens to apply. Anything (in this theme) that can be counted (because it is physical present and trivially discernable) finds a place in this list. These include similar objects (e.g. apples, people, leaves, etc.), categories (animals, plants, fruits, etc.), characteristics (colour, stage, shape, length, etc.), patterns & the like. Some entities lend themselves to counting only at a cognitive level or only in an amorphous sense and can be excluded. e.g. when considering the theme Our Ecosystem, kinds of pollution is one such entity. Children will also offer you countable entities which you might not have noticed. It would be best to work with them as when they come from the students, they will stick.
Lens 3 — Logical Operations:
This includes all the standard arithmetic operations (counting, addition, subtraction, multiplication, division, etc.) but also includes non-standard operations (skip counting, increments by 2, grouping by X, etc.) as well as matching (shapes, patterns, etc.). Students might come up with different operations as well and these should be admitted. Most importantly, these operations have to emerge from the theme. It should not be merely props from the theme but actual operations in the context of the theme. E.g. In the theme Fruits, “5 apples — 2 apples = ?” is an example of using fruits as a prop whereas “If you are supposed to eat 5 fruits per day & you have eaten 2 fruits during breakfast, how many more should you eat before you go to bed?” is an example of numerical operations rooted in the theme. Counting apples, counting oranges, counting bananas… become redundant after a point. Hence, simply listing all operations of counting/addition/etc. on the countable entities is not worth it. Strive to look for common tasks which are actually operations.
Lens 4 — Patterns:
A very important skill to develop is that of observing patterns. This develops their ability to observe deeper, develop a sense of cognitive rhythm as well as predict behaviour. It also helps them compare and contrast which manifests itself in the simplest form as Estimation. I, personally, hold estimation to be a very powerful and essential skill to be developed. To be able to estimate and perform back-of-the-envelope calculations is a very powerful skill to have in the adult world. To be able to set boundaries based on estimates gives us our “ballpark” figures. Apart from these, there arises a scope to develop aesthetic pattern recognition and this is useful in the fields of art, design and architecture. This lens helps us identify the various ways in which Art and Numeracy can be enmeshed.
Lens 5 — Extension Objects:
This turns out to be a bit of a tricky one. Extensions are essentially independent systems to which we can draw parallels from our current theme. E.g. If our theme is My Body, then an extension could be a machine because like our body, there are various specialised parts whose actions/functions need to be coordinated and requires maintenance. Most machines have an input (like our food, water, air) and produce an output (work, creativity, etc.) as well as waste (urine, excreta). If our theme is PYDS & My School, then we might draw parallels with a hospital in terms of infrastructure, experts, beneficiaries, salaries, processes, etc. It is important that an extension reveals multiple parallels which helps a student to derive abstractions, abstract operations as well as abstract data types. This helps them in constructing mental models and in designing solutions (because cross-pollinating models and abstractions is a key tool in solutioning).
Lens 6 — Stories:
Children learn best and in the stickiest manner when stories and songs are involved. That’s just how our brains are designed. Hence, the final lens allows us to construct stories using the real life actions (Lens 1) and filled with characters and objects (Lens 2) doing things (Lens 3) to create plots and themes and patterns (Lens 4). It would be great if the setting was in an extension to the theme (Lens 5) and in that way, we would have used all the 5 Lenses to help create stories and songs to fill Lens 6. This is singularly the biggest lacuna of the prescribed content (NCERT or XSEED or Cambridge or any other) — language is disconnected from numeracy and vice versa, and both proceed in independent directions.
Purpose of the Lenses:
The lenses are tools to help the teacher vivisect a theme and extract (to the fullest) mathematical contexts and phenomena. These lenses are for the teacher. The teacher could convert these into lessons and develop the same capability in the student. Philosophically, I believe it exposes our assumptions and blinders to our own, once-curious, self.
