I'm 48 and have gone through this process countless times, it feels like familiar rote:
1. Explore the subject almost randomly. Neither depth-first nor breadth-first; just follow threads that seem most useful or most interesting. Books, articles, videos, classes, whatever - it doesn't matter as long as you don't get bored. Dabble in all of them. If there are any practical skills involved (motion, writing code, etc) start doing those immediately.
2. At some point you get the ah-hah! moment. It is like the fog lifts and you can suddenly see the landscape. You don't know everything but now it seems like further knowledge is just a process of filling holes. You wield your knowledge to accomplish what you need by filling specific holes on-demand. Everything fits together like jigsaw puzzle pieces.
The time between #1 and #2 can be short or long depending on the complexity of the subject. But the experience of #2 is a massive dopamine rush. The main thing you need is patience; don't get frustrated and give up, you know that if you just keep poking around, eventually that dopamine rush will come.
I'm home-schooling my daughter in math because Zoom and it not agreeing with her. She did great last semester and tests several years ahead and has a very intuitive sense of why things work the way they do, I'd like to keep her interest.
However, we're now on to pre-algebra and using this book [0], "Prealgebra: The Art of Problem Solving". The first chapter is all about axioms, proofs of some sort, breaking down "obvious" conclusions back to their constituent proof-from-first-principles and it's not agreeing with her at all. Since this is the first week I'm struggling to find a way to have it all make sense, and I've concluded we need to kind of skip much of this first chapter (we'll look at the summary) and get to the later content which is more intuitive and applies "obvious" principles, and come back periodically to revisit the more mechanistic content in the first chapter. I think the parent post's description of exploring a topic matches how my daughter will come to understand the whole "algebra stack." (Wish me luck.)
I tutored algebra students and I reccommend the following approach with Khan academy/Openstax resources.
Order of Operations(No Variable)
Order of Operations(One Variable)
Advanced Order of Operations (One Variable)*optional
- Logs, Roots, Exponentials
- Polynomial multiplication/Factorization
Change the variable to random symbols
- Greek characters, shapes, animals
Order of Operations(Multiple Variables)
Introduce functions
- f(x) = mx + b
Connect Order of Operations with Functions
- Simplify an equation and plot as a function
Introduce units (basic physics equations)
- Simplify to formula and Plot
The goal in formal classes is to introduce order of operations with variables, the concept of functions, and how to manipulate those functions. If you are stem oriented, then this is the perfect time to introduce dimensional analysis(units) and physics-based functions. Instilling a sense of confidence and comfort with algebraic manipulation is critical. Prepping her for physics and working with functions is just smart.
Breaking things down to their basic axioms and rebuilding them with proofs is arguably quite an advanced approach. By Terry Tao's three stages of of rigour classification [0], this perspective isn't typical until undergraduate study.
It doesn't seem like a bad idea to give a taste of it significantly earlier, but I don't think it's surprising if someone without a more mature relationship with mathematics doesn't find much value in 'going back to basics.' If your daughter is doing well with a more intuitive approach it sounds like a good idea to stick with it for now.
Art of Problem Solving is great - I think they take credit for the US Math olympiad team's performance. I use books from the same organization at elementary school level and find them very good and fun to teach. I would recommend going a level lower among the books published by them [1] to see if that is a good starting point for your child. Their website also has some placement tests etc. to evaluate the right level for a student.
There's two hurdles for the pre-algebra student: numbers being generically represented as letters and the idea of axioms being generic rules for numbers. Check to see which issue or both is misunderstood. If it's numbers as letters, show them any interactive programming environment and demonstrate the idea of letters as generic numbers. From there, axioms are just more interactive demonstration.
I remember my mom struggling to teach me pre-algebra ahead of the school curriculum. My uncle (a math teacher) mailed us a bunch of "part-is-to-whole-as-part-is-to-whole" word problems of increasing complexity and various forms, and working through those made a lot of things click. The concreteness helped, and it also made me understand the equivalence between proportions, fractions, decimals, percents, and their real-world concepts, plus it has some simple solve-for-x attached to it. And it's learning-by-doing instead of learning-by-reading.
To expand on 1, I think it's been important for me to have some anchor points along the way. Some things I can "accomplish" however small they may be that, if nothing else, help me feel like I'm making progress I won't lose, even if full understanding takes much longer. Otherwise, backsliding in interest can mean the next time I get interested, I'm faced with the prospect of starting over from scratch, which can sometimes mean I just move on to something else.
Completely agree. I usually start with a trivial step and then build from it. Keeping notes on paper, in a markdown doc or jupyter notebook as appropriate is also useful. I have notes that I still look back at to remind me of details of areas I haven't used in a while.
The don't get bored part is really important. If it's not something you are required to learn, but instead you want to, then keep being interested and passionate is the best thing. There's no use in trying to optimize your learning at the very beginning if you drop it entirely in a month
It's fairly similar from what I've observed myself, but I'd say the ah-hah! moment for a given subject is not a singular event, but can be repeated if you look at the same subject from a different angle and get a different insight, deeper understanding of the solution or even of the problem. But maybe that's because learning in my case often is very much intertwined with trying to solve a problem.
And there's:
-1. Come across the solution by talking to people, reading books, articles, videos... long before actually needing it, sometimes not even being able to recall it in the next steps until the ah-hah! moment.
0. Have an actual need for that solution due to a problem you're trying to solve. And having that feeling of "there is a solution for this" but not being able to put your finger on it.
Talking to people helps a lot to speed up the process because everyone will have their own view on things which helps to shape your own understanding of the subject.
I agree with this. I would add that for me part of 1 is almost akin to pain. The frustration and annoyance I feel makes this part painful, but also when I get to 2 it's amazing. I hate feeling around in the dark but when the light comes on...
This resonates to my own experience! I'd only add that the step #2 (ah-hah! moments) are not singular, but milestones in a cycle. Being patient and pushing through #1 is definitely rewarding and expands to new fronts.
> But the experience of #2 is a massive dopamine rush. The main thing you need is patience; don't get frustrated and give up, you know that if you just keep poking around, eventually that dopamine rush will come.
I completely agree with you. I remember feeling the dopamine rush when I saw the 3b1b's video lectures on linear algebra and Andrew Ng's Machine Learning Lecture.
Although I strongly feel that simply reading books and watching video lectures are not the best way to learn on your own. You have to re-read them, use them, do the exercises and talk with peers to really get the underlying ideas. Maybe some better explainations exist, but hoping that you randomly stumble upon them as you browse through online articles is not a good idea, IMHO.
I like this quote from Prof. Michael Jordan:
"the first time you barely understand, the second time you start to get it, and the third time it all seems obvious."
You have to work with the material and come back to it again and again to really understand.
I have created a conversational learning medium, Primer, that is designed for self-learners. Our goal, right now, is to create conversational undergraduate-level computer science courses for anyone to learn on their.
It's primary focus is to bring resumability in play which books and video lectures lack. What I mean by that is that when you learn something from a book and video lectures, you start forgetting about it the minute you stop. After a month or so, your memory of the topics are maybe less than 30%. And if you haven't created flashcards or notes, then you have to skim over the book or watch the lectures to recollect.
On Primer, flashcards are automatically generated and your responses stored in the platform along with inline completions help to retrace & recover what you have learned. You can do the course for a while and resume again after quitting for a couple of months or years. Your responses act as memory breadcrumbs to help you retrace what you have learned.
You can test our two free courses over at https://primerlabs.io. ( No signup required ).
I have also created two comics-post about conversational learning that some of you may find useful.
In the end, you have to treat everything as something you will eventually come back and update. That's why I like networked-thoughts tool like Roam and obsidian, a little too much maybe.
This looks cool. Your product overlaps somewhat with Oppia (an open source tool for creating interactive 'explorations'). If you haven't seen it, I recommend you check it out.
Reminds me a bit of HyperCard. What is your interest in Oppia? I dabble in these things, but was thinking of working on some shared "decks" with my kid.
In the exploration I linked, the only types of interaction offered to the learner were either ok/proceed, or 'answer this textual multiple choice question'. This may make it seem like Oppia doesn't do much more than software for interactive fiction.
There are more similarity between things like Oppia and interactive fiction, I don't see them at odds. It is is wonderful that Oppia has a musical notation interface.
If you zoom out far enough, everything could be considered a quiz.
1. Explore the subject almost randomly. Neither depth-first nor breadth-first; just follow threads that seem most useful or most interesting. Books, articles, videos, classes, whatever - it doesn't matter as long as you don't get bored. Dabble in all of them. If there are any practical skills involved (motion, writing code, etc) start doing those immediately.
2. At some point you get the ah-hah! moment. It is like the fog lifts and you can suddenly see the landscape. You don't know everything but now it seems like further knowledge is just a process of filling holes. You wield your knowledge to accomplish what you need by filling specific holes on-demand. Everything fits together like jigsaw puzzle pieces.
The time between #1 and #2 can be short or long depending on the complexity of the subject. But the experience of #2 is a massive dopamine rush. The main thing you need is patience; don't get frustrated and give up, you know that if you just keep poking around, eventually that dopamine rush will come.