This is one reason why it's helpful to learn bottom-up when possible as opposed to diving straight into the deep end and trying to fill in missing knowledge as you go.
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Chief Quant, Director of Analytics at MathAcademy.com. Instead of optimizing return in the stock market, I optimize learning efficiency in students' brains.
I had fun writing them and I'm glad that they are making a positive impact, but since then I've been consumed by my work on Math Academy, which I find even more fun/impactful. (We do have a Methods of Proof course out, which is many times more scaffolded, refined, comprehensive, and generally instructionally superior to any textbook I could write independently, not to mention it's adaptive.)
So, long story short, I enjoyed writing those textbooks and am glad they're seeing the light of day, but I've moved on to a new chapter of life and don't plan on writing any more math books in the future (with the possible exception of something super niche like the math behind maximizing learning efficiency in hierarchical knowledge structures).
For instance, think about all the skills that a basketball player has to execute in parallel: they have to run around, dribble the basketball, and think about strategic plays, all at the same time. If they had to consciously think about the mechanics of running and dribbling, they would not be able to do both at the same time, and they would not have enough brainspace to think about strategy.
(This comment is basically the intro to a detailed article I wrote on the topic with plenty of scientific citations: https://www.justinmath.com/cognitive-science-of-learning-dev...)
I'm glad that you're having a good experience with the quizzes! That's exactly what we aim for: frequent, low-stakes, broad-coverage quizzes that challenge you to show off what you've learned (instead of stressing you out and overwhelming you). Used properly like this, quizzes are an extremely powerful learning tool. You get the benefits of retrieval practice, the spacing effect, and interleaving, all merged into one. And we pinpoint exactly what skills you need more practice on (which we have you review immediately afterwards), and by carefully calibrating the time constraint, we make sure you're developing your skills fully to the point of automaticity.
But, to clarify: the Eurisko book (Introduction to Algorithms and Machine Learning) only covers a small slice of the total roadmap from high school math to cutting-edge ML/AI. The other resources I recommend are the ones I mention in this post.
You can also get there from the Table of Contents -- to get there, go to your dashboard (where your learning tasks are), click on the name of your course in the top-left, and you'll be brought to a screen that says "Course Sequence" with a list of courses on the left and units on the right. If you expand the units, you'll see topics, and you can click on a topic to view its lesson in reference mode.
Note that XP & knowledge credit is only awarded for completing learning tasks on your dashboard, not for anything you do within a lesson in reference mode.
(But to answer your question: the math books on that page have "correct answer" solutions where you can tell if you got it right, but not fully-worked-out solutions. Introduction to Algorithms and Machine Learning technically does not have any solutions, but most of the problems involve constructing code implementations that match up with worked examples, or that give a desired result, so you can tell if you got it correct and in many cases you can follow along with the worked example to debug your code if it's not producing the desired output.)
But there's plenty more to learn in stages 2/3/4 before you reach the cutting edge. Would encourage you to read the article! :)
Should also point out that -- in the USA, at least -- linear algebra, multivariable calc, and calculus-based probability/statistics are not typically taught in high school. There's quite a bit of university-level math that someone needs to know if they want to reach cutting-edge ML/AI.
Sounds like you're well on your way to seriously leveling up your math skills, but if you have any questions about how the system works, how to get the most value out of it, the math learning process in general, etc., feel free to reach out anytime.
The Road to Excellence reads almost like a book club among researchers who read Bloom's Developing Talent in Young People and Ericsson's numerous papers on deliberate practice. Each chapter is from a different researcher or group, often with different views. (I was initially a bit put off by the lack of cohesiveness, but in hindsight, I like the resulting feel of authenticity: things get messy at the edge of human knowledge.)
And for anyone who hasn't read Developing Talent in Young People yet, I would highly recommend. The main idea is that Bloom studied the training backgrounds of 120 world-class talented individuals across 6 talent domains: piano, sculpting, swimming, tennis, math, & neurology, and what he discovered was that talent development occurs through a similar general process, no matter what talent domain. In other words, there is a "formula" for developing talent -- though executing it is a lot harder than simply understanding it. (More info: https://www.justinmath.com/book-review-bloom-developing-tale...)
I realize that tutoring is typically done at a lower frequency, but I don't agree that the lower frequency is ideal. At least in my mind, when I imagine the Platonic ideal of an education, there is no real difference between "lesson" and "homework" -- minimum effective doses of instruction are interspersed with minimum effective doses of active problem-solving, where every single problem is carefully selected in response to the learner's performance on the previous problem(s).
If this characterization of the Platonic ideal accurate, then achieving it would require a tutor continually sitting next to the student, analyzing their performance on every single problem that they do, and deciding the exact moment to move the student on to a new topic or problem type. Of course, that is infeasible with human tutors, so we settle for one or two days per week where the tutor tries to get the student prepared enough to tackle the homework without being completely overwhelmed.
I would argue that, while 1-2 tutoring sessions per week can really make a difference in a student's education, a lot of learning efficiency is still left unrealized (compared to 5 tutoring sessions per week).
See mathacademy.com/courses
But you're right that a free trial would produce more sign-ups and conversions, potentially enough to offset the additional support costs once we're in a better position to handle them (or once the product is better at preventing the need for them). Definitely something to think about in the future.
If you'd like to know more info about this, I went into depth in a StackExchange answer that blew up on HN a couple months ago: https://news.ycombinator.com/item?id=40348986
Normally you'd have to pay a tutor $50/hour. Multiply by 5 days a week, 52 weeks a year, you're at $13,000/year. Bringing that down to $499/year (26x cheaper) makes mathematical talent development accessible to many, many more people. Sure, that's not everyone, and there are still people who are priced out. But to me at least, providing a 26x cheaper option feels like a good starting point towards a goal of making mathematical talent development accessible to more and more people.
Of course, the following question still remains: "Why do you even need the learning experience to feel like working with a personal trainer? What is the benefit over a textbook, Khan Academy, MIT OpenCourseWare, etc.?"
The answer to that question: learning efficiency.
Let me tell you about my own experience. I self-studied a bunch of math subjects on MIT OpenCourseWare (OCW) when I was in high school. OCW is a good resource and I came a long way with it, but for the amount of effort that I put into learning on OCW, I could have gone a lot further if my time were used more efficiently. Just to name a handful of inefficiencies in OCW:
- not super scaffolded → you periodically run into situations where you bang your head on a wall thinking "how the heck did they get from here to there?" and it takes a long time to figure out what kind of logical leap is happening (if you figure it out at all)
- doesn't track your knowledge / make sure you've mastered the prerequisites for anything new you're supposed to learn → you often feel a large gap between your level of knowledge and the new material, which leads to more banging your head on a wall trying to figure out what prerequisite knowledge you're missing and how to learn it
- no spaced review → you quickly get rusty on a lot of what you learn, which not only means you come out of the course having forgotten a lot of content, but even during the course, you're constantly forgetting prerequisites
- doesn't adapt to your level of performance → you waste a lot of your time doing the wrong amount of work. Sometimes you grasp a topic quickly and end up doing way more practice problems than you need; other times you struggle with a topic and don't do enough practice problems to reach mastery
- leaves the definition of "mastery" open to interpretation by the learner → as a learner, it's hard to know when you've mastered something well enough to continue moving forward. You often think you've learned something well enough, when you actually haven't -- but you won't know unless there's an expert who is evaluating your knowledge. On the flipside, you can also take things too far being a perfectionist, spinning your wheels on the same topic for a week over some minor point that doesn't make perfect intuitive sense to you, when it would be more productive to just keep moving forward and solidify your understanding by building on top of it.
I could keep going with this list (happy to do so if you're interested, just let me know), but by now you probably get the point: all of these things introduce unproductive friction into the learning process, leading you to make less educational progress per unit time/effort that you put towards learning.
That's one reason why I've been so motivated to help build Math Academy. We take away as much of this learning friction as possible and maximize your learning efficiency. We address all the above issues and more.
That's our main value proposition: sure, it's possible to learn math elsewhere, but it's way more efficient with us.
Efficiency is important not only because you make faster progress, but also because you're less likely to quit.
In practice, people get off the train and stop learning math once it begins to feel too inefficient. In anything you do, once the progress-to-work ratio gets too low, you're going to lose interest and focus on other endeavors where your progress-to-work ratio is higher.
Efficiency keeps that progress-to-work ratio as high as possible, keeping you on the math learning train as long as possible.
> 30-Day Money Back Guarantee. We're so confident that Math Academy will help you or your child master advanced math concepts faster than any other method on the planet that if you find it doesn't suit your needs within the first month, we will refund your payment. There are no long term contracts, no additional textbooks to buy, and you may cancel your membership at anytime in the future.
Yeah, if you've got a deep hole in your knowledge, it can be difficult to know where the bottom is (which is problematic because that's where you have to start filling it in).
1. Our Mathematics for Machine Learning course: https://www.mathacademy.com/courses/mathematics-for-machine-...
2. We designed a Mathematical Foundations course sequence specifically for adults who want to get up to speed or relearn math skills they have forgotten as preparation for Mathematics for Machine Learning and other university math courses. More info here: https://www.mathacademy.com/adult-students.
3. When you start on the system we assess not only your knowledge of the course you're placing into, but also lower-level foundational topics. If we detect that you have any gaps in your foundations (most learners do, especially adults who have been out of the game for a while), then we'll automatically add them to your learning plan and make sure you learn them before we give you any more advanced topics that depend on them. More info here: https://www.mathacademy.com/how-our-ai-works#diagnostic-algo...
Spacing is one example of a desirable difficulty; the spacing effect has been known for over a century and it has been replicated so many times since then that it might as well be a law of physics. https://en.wikipedia.org/wiki/Spacing_effect
That's totally fine and all it means is you may need to start off in a lower course to shore up your missing foundations [1].
There have been so many people in this situation that we actually designed a Mathematical Foundations course sequence specifically for adults who want to get up to speed or relearn math skills they have forgotten (from fractions through calculus) as preparation for upper-level university math courses. More info here: https://www.mathacademy.com/adult-students.
Please do let me know if you have any follow-up questions about that or if anything is unclear. I'm always happy to chat with people who are serious about learning math.
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Footnotes
[1] Note that we do check for missing foundations during diagnostics, and any that we find we'll add to your learning plan -- but currently there's a limit to how far we look back. For Linear Algebra, we only look back to the beginning of Algebra 2. So if you're rusty on any Algebra 1 stuff -- factoring, quadratic equations, systems of linear equations, etc. -- or arithmetic stuff like working with fractions/exponents, then you'll need to drop back to a lower course to shore up those foundations.
> any plans on expanding into physics problems? would LOVE these ideas to delve back into phsyics. (especially circuits.)
Our grand plan is to completely fill out out math courses, then expand to other related fields such as computer science and physics.
Math Academy is built with a different context in mind:
-- 1. we have tons of different topics (for instance, over 300 in our AP Calculus BC course -- and that's just one course; students who stick with us past their first course and continue taking more courses on our system can easily accumulate thousands of topics on which they need to maintain their knowledge)
-- 2. each repetition consists of multiple questions on a topic (after an initial lesson task consisting of somewhere in the ballpark of 10 questions, future repetitions are review tasks in the range of 3-5 review questions)
-- 3. students are getting through a new topic every 20 minutes or so on average (our AP Calculus BC course is estimated to take about 6000 minutes, and that includes time spent on review / quizzes / etc., and 6000 minutes / 300 topics = 20 minutes/topic)
Just some back-of-the envelope math in our context: say you learn 3 new topics per day (an hour-long session), and each review takes you about 4 questions. Then, as a rough estimate, pretty quickly you reach a point where you've got
12 review questions based on yesterday's lessons,
+ another 12 review questions on topics you reviewed for the first time last week,
+ another 12 review questions on topics you reviewed for the second time a couple weeks ago, + another 12 review questions on topics you reviewed for the third time a month ago,
+ another 12 review questions on topics you reviewed for the fourth time a couple months ago,
...
That's already 60 review questions/day, already past the point where you’re spending every hour-long session entirely on review, which means that your progress grinds to a halt in terms of learning new material.
So, in our context at least, raw spaced repetition just doesn't work out in a way that’s feasible – student will get absolutely crushed by a tsunami of review unless we unless we take measures to drastically cut down the amount of review (i.e., fractional implicit repetition + repetition compression).
I hear your point about needing some refresher if you take 2-3 months off and come back, though. Similar thing happens with Math Academy students if they take time off and come back, but the way we solve that problem is we just recommend they take a new diagnostic to refresh their knowledge profile. Basically, it just peels back their "knowledge frontier" to a point where they can pick up and continue learning smoothly with our standard approach.
(In my experience, that’s what you’d ideally do as a teacher after summer vacation – you know your students have forgotten a lot of material, so you have them take a beginning-of-the-year knowledge evaluation and go from there.)