Please note this will be a very long post with many examples and references , and I will keep updating it .
So I want to talk about a new idea, That I discovered after realizing the limitations of books , Lets Start this by first defining what a book is xp
I know you are wondering why do we need a definition of a book? Well I just want to emphasize on few key points and limtations .
Books are Published content that authors write to communicate with their target audience which are based on certain assumptions .
That assumptions word is very important here
- Book Doesnt Teach How to “use/read” a book ? Why is this important? Well it might sounds very simple and stupid but every book has certain basic assumptions that they expect from readers/users. Imagine Starting over with chapter 5 from a algebra book , which has a chapter on quadratic eqations, while most books will mention the dependencies of that chapter some badly designed books wont .
2. Books assume readers have good memories (especially maths textbooks) , Open up a chapter on Trigonometry or Properties of Triangles and just look at the “Ocean” of formulas they expect us to “remember” , While some brave souls might remember all of them , most of like me dont .
3. Books Dont check if what you have “read” you have actually practiced that when did you practice that, Books Dont provide you powerful analytics .
4. Books dont help you realize patterns and dependencies among different entities.
5. Books dont give you the intuition that is very human friendly (they are very static ) , It can be the case that ever since the book was published or even before it was published there were other intuitive explanations of a particular concept that were way simpler that what book provided but the author never cared to research over those before writing the book.
Even though we have a lot of limitations Books still remain one of the widely used resource to “learn” concepts . They are cheap and widely available.
Alrightly , In this post I want to specifically magnify upon the 4th point and why I have realized its so important (atleast from mathematical perspective ) .
Imagine how hard it is to just rote learn all those hard formulae, but thats we are trained to do , I reject this way of learning things because of the following reasons :
- Not every student/person has a capacity to store a large number of formulae in his/her LTM (Long Term Memory)
- Jumping directly to formulae can eliminate intuition and reasoning
- Formulae are limited and most often we need to chain them in order to solve a problem , remembering the exact sequence is also challenging.
So How Can we solve this problem ? I introduce ” The Toolkit ” , It’s a very simple abstract model which you can think of as “virtual memory” .
Toolkit enumerates all the rules (not necessarily formulae ) that a student has learnt . But here is the catch , It also includes “Patterns” that books fail to realize. Each Rule/ Pattern has a Index number , Title , Description , Date Added Fields
We can either make use of a diary to make our “Toolkit” or just use any text editing software like notepad, Thats what toolkit model really is , Its just text .
Why are patterns so useful , Lets take up an example in properties of triangles unit.
My Book has 3 formulae :
sin(A) = area / 2bc , and similarily quivalent versions for sin(B) and sin(C)
Now , I am sure most of you must have simply “learnt” it and stored it in your LTM ( Thats what we assume ) . Imagine that after 1 year you changed your major and started learning “Cooking” , and around that time I give you a problem that heavily depends on these formulae , will you able to solve it ?
Here is How Our Toolkit Model solves this problem , We first proof this to convince our selves and notice that the proof takes less than 1 minute . We realize a pattern/rule of the following form :
Description : For relating Sine < A > to area of triangle => Consider general triangle => Draw perpendicular => Apply law of sines in the subtriangle .
Thats It. You know whats nice ? The first time you did this on your own , It provided you confidence that It is not that hard and simpler that directly rote learning the formula. Try the same problem after 1 year referring to your “Toolkit” I am sure you will solve it.
Some things related to Patterns/Rules :
The “Toolkit” doesnt enforce any constraints on writing style/form of productions , they depend on the person who is going to “use” his/her “toolkit”
The entries/rules in your toolkit cannot be proved but are based on your “assumptions” , Thats why you should only add entries when you first make sure you are confident deriving them. The whole point of introducing toolkit is to remove burden of memorizing “everything” .
The Toolkit is dynamic in nature and you must expand entries , You can even generalize rules/patterns.
Here is one practical use case of toolkit concept , So I was solving some problems, I noticed that many problems were strikingly similar in nature, and there was a “pattern” in them that seemed recursive, I couldnt find any specific formulae to solve these kinds of problems that were simple .
So Here is What I did I solved the first problem , It gave me the confidence , I continued solving similar problems and identified the pattern , You can watch these in this videos -> https://www.youtube.com/watch?v=DHGgmgAeJ_o
Then I added another entry in my toolkit , which looks something like this :
Description: Relating Sides of a triangle with tan/cot of half angle -> Law of Sines ->Trig Transformations -> Result 🙂
Just look at the above entry that I added in my toolkit , It is so personal , It is so informal , and It gives me the confidence to solve any problem of similar nature. In fact I solved more problems by applying rule 2 of my toolkit and reached out at a correct solutions.
User have to firmly believe in the entries added in their toolkit .It may rarely occur that you added a slightly misleading / incorrect rule , You must first challenge the toolkit , If you fail , you must modify/remove your entry.
The feature image of this post shows one example of my toolkit entry
Toolkit Model Directly Endorses / Appreciates The wonderful concept of Constructionism because Users must solve the problems on their own , Get the confidence , Identify Patterns and Store that very confidence permanently in form of Toolkit entry .